On Sylow 2-Subgroups of Finite Simple Groups of Order up to 210

Home , Extra special group

ON SYLOW 2-SUBGROUPS OF FINITE SIMPLE GROUPS OF ORDER UP TO 210

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

Sergey Malyushitsky, M.S.

*****

The Ohio State University

2004

Dissertation Committee: Approved by

Professor Koichiro Harada, Adviser Professor Akos Seress Adviser Professor Ronald Solomon Department of Mathematics ABSTRACT

A 2-group ( a group of order a power of 2 ) is called realizable if it occurs as a

Sylow 2-subgroup of a ﬁnite simple group. The purpose of this thesis is to study all

realizable groups of order up to 210. From the classiﬁcation of all simple groups of

finite order we know all realizable groups of order up to 210 as Sylow 2-subgroups of known finite simple groups. However without the use of classification determining all realizable 2-groups is very difficult.

In the ﬁrst part of the thesis we present an argument that produces all realizable groups of order up to 32, by eliminating one by one all 2-groups that can not occur as a Sylow 2-subgroup of a ﬁnite simple group. When the number of 2-groups of given order becomes too large to handle it by hand we attempt to use a computer for repetitive checks on a large number of 2-groups.

The second part of the thesis is devoted to describing all realizable 2-groups of order up to 210 using the classification of all finite simple groups. We determine the identification number of each group in the Small Groups Library in GAP4 and compute the power-commutator presentation of each realizable group S of type G.

We also determine the conjugacy classes of involutions of S and their fusion in G, maximal subgroups of S, maximal abelian and maximal elementary abelian subgroups of S, Z(S), S/Z(S), maximal quotient groups of S and maximal normal extra-special subgroups of S.

ii This is dedicated to my family.

iii ACKNOWLEDGMENTS

I would like to express my deep gratitude to my adviser, Professor Koichiro

Harada, for his guidance and support.

I would like to thank Professor Akos Seress for his advice on various problems related to GAP4.

I would also like to thank Professor Ronald Solomon and Professor Sia Wong for their comments and suggestions.

And ﬁnally, I would like to thank my family, and especially my wife Svetlana, for their love and support.

iv VITA

1972 ...... Born - Minsk, Belarus

1997 ...... M.S. Mathematics

2000 ...... M.S. Computer Science

1993-2001 ...... Graduate Teaching Associate, The Ohio State University. 2001-2002 ...... Software Engineer, IMAKE Software & Services, Inc. 2002-present ...... Graduate Teaching Associate, The Ohio State University.

FIELDS OF STUDY

Major Field: Mathematics

v TABLE OF CONTENTS

Page

Abstract ...... ii

Dedication ...... iii

Acknowledgments ...... iv

Vita...... v

List of Tables ...... ix

List of Figures ...... x

Chapters:

1. Introduction ...... 1

2. Quoted results and consequences ...... 11

2.1 Notation ...... 11 2.2 Quoted results ...... 12 2.3 Main consequences ...... 14

3. Determining realizable 2-groups ...... 18

3.1 2-groups of order ≤ 8...... 18 3.2 2-groups of order 16 ...... 19 3.3 2-groups of order 32 ...... 22 3.4 2-groups of order 64 ...... 30

vi 4. Properties of Realizable groups of order 26 ...... 37

4.1 Realizable group of type L2(64) ...... 37 6 7 4.2 Realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ) ...... 38 4 5 4 4.3 Realizable group of type L3(q), q ≡ 2 − 1(mod 2 ),U3(q), q ≡ 2 + 1(mod 25)...... 40 4.4 Realizable group of type Sz(8) ...... 42

4.5 Realizable group of type U3(4) ...... 43 2 4.6 Realizable group of type A8,A9,L4(2),U4(2),S4(q), q ≡ 2 ± 1(mod 23)...... 44

4.7 Realizable group of type L3(4) ...... 46 2 3 4.8 Realizable group of type M12, D4(q), q ≡ 2 ± 1(mod 2 ), G2(q), q ≡ 22 ± 1(mod 23)...... 47

5. Realizable Groups of Order 27 ...... 50

5.1 Realizable group of type L2(128) ...... 50 7 8 5.2 Realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ) ...... 51 5 6 5.3 Realizable group of type L3(q), q ≡ 2 − 1(mod 2 ) and U3(q), q ≡ 25 + 1(mod 26)...... 53 3 4 3 5.4 Realizable group of type L3(q), q ≡ 2 + 1(mod 2 ),U3(q), q ≡ 2 − 1(mod 24)...... 54

5.5 Realizable group of type A10 ...... 56 3 5.6 Realizable group of type M22, M23, McL, L4(q), q ≡ 5(mod 2 ), 3 U4(q), q ≡ 3(mod 2 )...... 58 5.7 Realizable group of type J2, J3 ...... 60

6. Realizable Groups of Order 28 ...... 63

6.1 Realizable group of type L2(256) ...... 63 6.2 Realizable group of type S4(4) ...... 65 8 9 6.3 Realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ) ...... 67 6 7 6 6.4 Realizable group of type L3(q), q ≡ 2 − 1(mod 2 ),U3(q), q ≡ 2 + 1(mod 27)...... 69 3 4 3 6.5 Realizable group of type G2(q), q ≡ 2 ± 1(mod 2 ),D4(q), q ≡ 2 ± 1(mod 24)...... 71 3 4 6.6 Realizable group of type S4(q), q ≡ 2 ± 1(mod 2 ) ...... 72 6.7 Realizable group of type Ly ...... 75

7. Realizable Groups of Order 29 ...... 77

7.1 Realizable group of type L2(512) ...... 77

vii 9 10 7.2 Realizable group of type L2(q), q ≡ 2 ± 1(mod 2 )...... 79 7 8 7 7.3 Realizable group of type L3(q), q ≡ 2 − 1(mod 2 ),U3(q), q ≡ 2 + 1(mod 28)...... 80 4 5 4 7.4 Realizable group of type L3(q), q ≡ 2 + 1(mod 2 ),U3(q), q ≡ 2 − 1(mod 25)...... 81 4 4 7.5 Realizable group of type L4(q), q ≡ 7(mod 2 ),U4(q), q ≡ 9(mod 2 ) 83 3 3 7.6 Realizable group of type L5(q), q ≡ 3(mod 2 ),U5(q), q ≡ 5(mod 2 ) 85 2 3 7.7 Realizable group of type S6(q), q ≡ 2 ± 1(mod 2 ) ...... 87 2 3 7.8 Realizable group of type S6(2),A12,A13,O7(q), q ≡ 2 ± 1(mod 2 ) . 90 7.9 Realizable group of type L3(8) ...... 93 7.10 Realizable group of type U3(8) ...... 95 7.11 Realizable group of type HS ...... 96 7.12 Realizable group of type O0Nan ...... 98

8. Realizable Groups of Order 210 ...... 100

8.1 Realizable group of type L2(1024) ...... 100 10 11 8.2 Realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ) ...... 102 8 9 8 8.3 Realizable group of type L3(q), q ≡ 2 − 1(mod 2 ),U3(q), q ≡ 2 + 1(mod 29)...... 103 4 5 4 8.4 Realizable group of type D4(q), q ≡ 2 ± 1(mod 2 ),G2(q), q ≡ 2 ± 1(mod 25)...... 105 4 5 8.5 Realizable group of type S4(q), q ≡ 2 ± 1(mod 2 ) ...... 106 8.6 Realizable group of type He, M24,L5(2) ...... 108 8.7 Realizable group of type Co3 ...... 111 4 4 8.8 Realizable group of type L4(q), q ≡ 9(mod 2 ),U4(q), q ≡ 7(mod 2 ) 113 8.9 Realizable group of type U5(2) ...... 116 8.10 Realizable group of type Sz(32) ...... 118 − 2 3 8.11 Realizable group of type A14,A15,O8 (q), q ≡ 2 ± 3(mod 2 ) . . . 120

Bibliography ...... 124

viii LIST OF TABLES

Table Page

1.1 Realizable 2-groups of order 22 and their types...... 1

1.2 Realizable 2-groups of order 23 and their types...... 2

1.3 Realizable 2-groups of order 24 and their types...... 2

1.4 Realizable 2-groups of order 25 and their types...... 2

1.5 Realizable 2-groups of order 26 and their types...... 3

1.6 Realizable 2-groups of order 27 and their types...... 3

1.7 Realizable 2-groups of order 28 and their types...... 4

1.8 Realizable 2-groups of order 29 and their types. (For groups of type

L3(8) and U3(8) the ID number identifying them in the Small Groups Library in GAP4 is not computed as the number of 2-groups of the same order, rank and p-class is too high to complete the comprehensive search for these groups.) ...... 4

1.9 Realizable 2-groups of order 210 and their types. (The group ID number is not given because the 2-groups of order 210 are not enumerated in the Small Groups Library in GAP4.) ...... 5

1.10 The table shows the number of realizable 2-groups of given order along with the total number of 2-groups of the same order and also the number of 2-groups of class 2...... 5

ix LIST OF FIGURES

Figure Page

1.1 Realizable groups inclusion tree...... 10

x CHAPTER 1

INTRODUCTION

The main purpose of the dissertation is to investigate the internal structure of realizable 2-groups of order up to 210 and describe the most interesting internal properties of such groups. A group S is called realizable if it occurs as a Sylow-2 subgroup of a ﬁnite simple group G. We then call S of type G.

Using the full classiﬁcation of simple groups, see [7], one can list all realizable 2-groups of order ≤ 210. (see Tables 1.1 - 1.9)

Each table shows all realizable 2-groups of given order. We also added the ID number identifying each group in the Small Groups Library in GAP4 established by Bettina

Eick and E. O’Brien (see [12], [13], [3], [2]).

From Table 1.10 one can see that compared to the total number of 2-groups of given order the number of realizable 2-groups of that order is very small. A natural

Realizable groups of order 4 Group Number Group description Group of type 2 3 [4,2] E4 L2(4),L2(q), q ≡ 2 ± 1(mod 2 )

Table 1.1: Realizable 2-groups of order 22 and their types.

1 Realizable groups of order 8 Group Number Group description Group of type 2m+1 [8, 5] E8 J1,L2(8),R(3 ) [8, 3] D8 A6,A7,L3(2), 3 4 L2(q), q ≡ 2 ± 1(mod 2 )

Table 1.2: Realizable 2-groups of order 23 and their types.

Realizable groups of order 16 Group Number Group description Group of type [16, 14] E16 L2(16) 4 5 [16, 7] D16 L2(q), q ≡ 2 ± 1(mod 2 ) 3 [16, 8] SD16 M11,L3(q), q ≡ 3(mod 2 ), 3 U3(q), q ≡ 5(mod 2 )

Table 1.3: Realizable 2-groups of order 24 and their types.

Realizable groups of order 32 Group Number Group description Group of type [32, 51] E32 L2(32) 5 6 [32, 18] D32 L2(q), q ≡ 2 ± 1(mod 2 ) 3 4 [32, 19] SD32 L3(q), q ≡ 2 − 1(mod 2 ), 3 4 U3(q), q ≡ 2 + 1(mod 2 ) 2 3 [32, 11] 4 o 2 L3(q), q ≡ 2 + 1(mod 2 ), 2 3 U3(q), q ≡ 2 − 1(mod 2 )

Table 1.4: Realizable 2-groups of order 25 and their types.

2 Realizable groups of order 64 Group Number Group description Group of type [64, 267] E64 L2(64) 6 7 [64, 52] D64 L2(q), q ≡ 2 ± 1(mod 2 ) 4 5 [64, 53] SD64 L3(q), q ≡ 2 − 1(mod 2 ), 4 5 U3(q), q ≡ 2 + 1(mod 2 ) [64, 82] 23+3 Sz(8) 2+4 [64, 245] 2 U3(4) 3 [64, 138] 2 : D8 L4(2),A8,A9,U4(2) 2 3 S4(q), q ≡ 2 ± 1(mod 2 ) 4 2 [64, 242] 2 : 2 L3(4) 2 2 2 3 [64, 134] 4 : 2 M12,D4(q), q ≡ 2 ± 1(mod 2 ), 2 3 G2(q), q ≡ 2 ± 1(mod 2 )

Table 1.5: Realizable 2-groups of order 26 and their types.

Realizable groups of order 128 Group Number Group description Group of type [128, 2328] E128 L2(128) 7 8 [128, 162] D128 L2(q), q ≡ 2 ± 1(mod 2 ) 5 6 [128, 163] SD128 L3(q), q ≡ 2 − 1(mod 2 ), 5 6 U3(q), q ≡ 2 + 1(mod 2 ) 3 4 [128, 67] 8 o 2 L3(q), q ≡ 2 + 1(mod 2 ), 3 4 U3(q), q ≡ 2 − 1(mod 2 ) 3 [128, 928] D8 o 2 A10,A11,L4(q), q ≡ 3(mod 2 ), 3 U4(q), q ≡ 5(mod 2 ) 4 3 [128, 931] 2 : D8 M22,M23, McL, U4(q), q ≡ 3(mod 2 ), 3 L4(q), q ≡ 5(mod 2 ) 2+4 [128, 934] 2 : 2 J2,J3

Table 1.6: Realizable 2-groups of order 27 and their types.

3 Realizable groups of order 256 Group Number Group description Group of type [256, 56092] E256 L2(256) 6 2 [256, 8935] 2 : 2 S4(4) 8 9 [256, 539] D256 L2(q), q ≡ 2 ± 1(mod 2 ) 6 7 [256, 540] SD256 L3(q), q ≡ 2 − 1(mod 2 ), 6 7 U3(q), q ≡ 2 + 1(mod 2 ) 3 4 [256, 5298] G2(7) G2(q), q ≡ 2 ± 1(mod 2 ), 3 4 D4(q), q ≡ 2 ± 1(mod 2 ) 3 4 [256, 6661] S4(7) S4(q), q ≡ 2 ± 1(mod 2 ), [256, 6665] 2(2 o 2 o 2) Ly

Table 1.7: Realizable 2-groups of order 28 and their types.

Realizable groups of order 512 Group Number Group description Group of type [512, 10494213] E512 L2(512) 9 10 [512, 2042] D512 L2(q), q ≡ 2 ± 1(mod 2 ) 7 8 [512, 2043] SD512 L3(q), q ≡ 2 − 1(mod 2 ), 7 8 U3(q), q ≡ 2 + 1(mod 2 ) 4 5 [512, 947] 16 o 2 L3(q), q ≡ 2 + 1(mod 2 ), 4 5 U3(q), q ≡ 2 − 1(mod 2 ) 4 [512, 60809] D24 o 2 L4(q), q ≡ 7(mod 2 ), 4 U4(q), q ≡ 9(mod 2 ) 3 [512, 60833] SD24 o 2 L5(q), q ≡ 3(mod 2 ), 3 U5(q), q ≡ 5(mod 2 ) 2 3 [512, 7530110] S6(3) S6(q), q ≡ 2 ± 1(mod 2 ) 6 [512, 406983] 2 : D8 A12,A13,S6(2), 2 3 O7(q), q ≡ 2 ± 1(mod 2 ) 6 3 [512, ?] 2 : 2 L3(8) 3+6 [512, ?] 2 U3(8) 3 [512, 60329] 4 : D8 HS 3 0 [512, 58362] 4 D8 O Nan

9 Table 1.8: Realizable 2-groups of order 2 and their types. (For groups of type L3(8) and U3(8) the ID number identifying them in the Small Groups Library in GAP4 is not computed as the number of 2-groups of the same order, rank and p-class is too high to complete the comprehensive search for these groups.)

4 Realizable groups of order 1024 Group description Group of type E1024 L2(512) 10 11 D1024 L2(q), q ≡ 2 ± 1(mod 2 ) 8 9 SD1024 L3(q), q ≡ 2 − 1(mod 2 ), 8 9 U3(q), q ≡ 2 + 1(mod 2 ) 4 5 G2(17) G2(q), q ≡ 2 ± 1(mod 2 ), 4 5 D4(q), q ≡ 2 ± 1(mod 2 ) 4 5 S4(17) S4(q), q ≡ 2 ± 1(mod 2 ) 6 2 :(D8 × 2) He, M24,L5(2) 4 3 2 (2 : D8) Co3 4 L4(9) L4(q), q ≡ 9(mod 2 ), 4 U4(q), q ≡ 7(mod 2 ) 4+4 2 2 : 2 U5(2) 25+5 Sz(25) − (D8 × D8) o 2 A14,A15,O 8(q), q ≡ 3(mod 8)

Table 1.9: Realizable 2-groups of order 210 and their types. (The group ID number is not given because the 2-groups of order 210 are not enumerated in the Small Groups Library in GAP4.)

Realizable groups statistics Group Order Number of Realizable Total Number class ≤ 2 percentage 2 0 1 1 100 4 1 2 1 100 8 2 5 1 100 16 3 14 11 79 32 4 51 33 65 64 8 267 128 48 128 7 2328 962 41 256 7 56092 31764 56 512 12 10494213 8843770 84 1024 11 49487365422 ? ?

Table 1.10: The table shows the number of realizable 2-groups of given order along with the total number of 2-groups of the same order and also the number of 2-groups of class 2.

5 question arises of whether one can describe the internal properties of realizable 2-

groups that will distinguish them apart from all other 2-groups of the same order. K.

Harada and M. L. Lang proposed the following problems associated with realizable

2-groups, see [11]:

• (N) Classify all 2-groups that are realizable without using the classiﬁcation of

all simple groups.

• (P) Describe all 2-groups that are realizable.

• (E) Determine all realizable 2-groups of order at most 246.

This dissertation consists of two major parts. In part one in Chapters 2 and 3 we solve problem (N) for 2-groups of order ≤ 25 and use the methods developed to eliminate about 63% of non-realizable 2-groups of order 26. It should be noted that without the restriction on the group order problem (N) is extremely diﬃcult. In Chapter 2 we prove some propositions that we use to conclude that a given 2-group can not occur as a Sylow 2-subgroup of a non-abelian simple group, so is not realizable. In Chapter 3 we produce all realizable 2-groups (without the use of classiﬁcation of simple groups) by going through the list of 2-groups of given order and eliminating all 2-groups that can not be realizable. For groups of order 32 the proof relies on the the list of 2- groups obtained by M. Hall and J. K. Senior (see [8]). The number of 2-groups of order 64 is too large to handle it by hand, so we use an algorithm described at the end of Chapter 3 and implemented in GAP4 to eliminate about 63% of non-realizable

2-groups of order 64.

The second part of the dissertation is devoted to solving problem (P) for realizable

2-groups of orders from 26 to 210.

6 There are 55 realizable 2-groups of order ≤ 210 in total. In Chapters 4 - 8 we present

the following information on each realizable 2-group S starting with groups of order

26 and up to 210:

We ﬁnd

• Number identifying each group in the Small Groups Library in GAP4

• Generators and relations of S

• Z(S) and S/Z(S)

• Maximal subgroups of S. (We also check whether S/Z(S) is isomorphic to any ∼ of the maximal subgroups of S if Z(S) = Z2).

• S/Φ(S), (where Φ(S) is Frattini Subgroup of S)

• Maximal quotient groups of S

• Conjugacy classes of maximal abelian subgroups A of S and their fusion in G,

as well as NG(A)/CG(A) (where S ∈ Syl2(G) and G is a simple group)

• Conjugacy classes of maximal elementary abelian subgroups A of S and their

fusion in G, as well as NG(A)/CG(A) (where S ∈ Syl2(G) and G is a simple

group)

• Conjugacy classes of maximal normal extra-special subgroups E of S and their

fusion in G, as well as NG(E)/CG(E) (where S ∈ Syl2(G) and G is a simple

group)

• Conjugacy classes of involutions of S and their fusion in G (where S ∈ Syl2(G)

and G is a simple group)

7 We use computer algebra system GAP4 (see [14]) for most calculations. The algorithms to compute the corresponding properties are implemented using GAP4 - commands. For groups of orders up to 28 we use GAP4 - command ”IdSmallGroup”,

(c.f. [14]) to identify the group S and its’ various subgroups by their ID number in the Small Groups Library in GAP4. However for groups of order 29, which are enumerated in the library, this command is not available. In order to determine the

ID number of the group we have to use some group invariants and perform a comprehensive search among all 2-groups of the same order, rank and p-class. In most cases using Z(S), S/Z(S), number of elements of different orders and maximal subgroups of a 2-group S as invariants allows us to identify the group ID number uniquely. How- ever for rare cases when there is more than one 2-group with the same invariants we have to broaden the list of invariants and also include the number of elements in each conjugacy class and the order of Aut(S). The same approach is used to determine whether different 2-groups are isomorphic. Although in GAP4 there are a number of commands available to determine whether two groups are isomorphic they rarely bring results in reasonable time. We find a group ID number for each group in Small

Groups Library in GAP4 instead and compare the ID numbers to determine the isomorphism.

We observed a certain ”duality” for some of the realizable 2-groups investigated. In ∼ particular if Z(S) = Z2 it happens for some groups S that S/Z(S) is a realizable

2-group of lower order and is isomorphic to one or more maximal subgroups of S.

This information is represented in the following ﬁgure (see (1.1)). The groups of the same order are arranged in columns. We label the vertices of the resulting graph by realizable 2-groups. Elementary abelian, dihedral and semidihedral groups are

8 omitted. Other realizable 2-groups are labeled by their type. A vector connects 2 vertices labeled S1 and S2 if a realizable 2-group of lower order S1 occurs among maximal subgroups of a realizable group of higher order S2, a vector connects 2 vertices ∼ labeled S2 and S1 if S1 = S2/Z(S2). Multiple vectors on the graph connecting the same groups S1 and S2 represent the case when multiple maximal subgroups of S2 are isomorphic to S1. There is one special case when we observe the ”duality” for groups

S1 and S2 but group S1 is not realizable. It occurs for S2 of type L5(3). There exists ∼ ∼ S1 = [256, 6662] which is not realizable but S1 = S2/Z(S2) and there is one maximal subgroup of S2 isomorphic to S1.

2 3 It also occurs for S2 of type A8,A9,L4(2),U4(2),S4(q), q ≡ 2 ± 1(mod 2 ). There ex- ∼ ∼ ists S1 = [32, 27] which is not realizable but S1 = S2/Z(S2) and there is one maximal subgroup of S2 isomorphic to S1.

9 - - A - M Ly L (17) G (17) 8 22 3 2 @@I @I@ @ @@ @ © - - - L (4) @ @@R A - S (7) - L (7) - S (17) 3 @ @@@R 10 4 4 4 @ @ @ @ - M @@R J A - A 12 2 12 - 14 @I@ @ U (4) L (9) - G (7) S (3) @ Co 3 3 2 6 @ 3

Sz(8) S4(4) L5(3) L4(9)

L3(8) Sz(32)

U3(8) U5(2)

HS M24

O0Nan

Figure 1.1: Realizable groups inclusion tree.

10 CHAPTER 2

QUOTED RESULTS AND CONSEQUENCES

2.1 Notation

Notation 2.1.1 Let us introduce some notation. Most of it is standard.

Sylp(G) - a set of Sylow-p subgroups of G.

G0 - the commutator subgroup of G.

O2(G) - the maximal normal 2-subgroup of G.

O20 (G) - the maximal normal subgroup of odd order of G.

O20 ,2(G) - the preimage of O2(G/O20 (G)) in G. Z(G) - the center of G.

∗ Z (G) - the preimage of Z(G/O20 (G)) in G. Φ(G) - the Frattini subgroup of G.

i Ipi (P ) - the set of elements of order p of a p-group P .

i Ωi(P ) - the subgroup of a p-group P generated by the elements of order p .

G# - the set of non-identity elements of G.

< a, b, ...| · · · > - the power-commutator presentation of the group.

Zn - a cyclic group of order n.

Dn - a dihedral group of order n.

SDn - a semi-dihedral group of order n.

11 Qn - a generalized quaternion group of order n.

|M| - the cardinality of the set M.

|M|2 - the maximal 2-factor of |M| . H < G - H is a proper subgroup of G.

H/G - H is a normal subgroup of G. x ∼G y - an element x is conjugate with y in G. tg = g−1 · t · g. tG = {tg|g ∈ G}.

S = [2n, k] - k-th group of order 2n in the Small Group Library in GAP 4.

2.2 Quoted results

In this section for completeness we quote the results that are used in the following sections.

Theorem 2.2.1 Let G be a ﬁnite group of order divisible by prime number p and P

∈ Sylp(G). If x, y ∈ Z(P ) are conjugate in G then they are conjugate in NG(P ).

Theorem 2.2.2 (Glauberman Z∗-Theorem (see [4])) Let G be a ﬁnite group of

G even order and S ∈ Syl2(G). Assume t is any non-identity element of G. If t ∩ S = t then G = O20 (G) · CG(t)

Theorem 2.2.3 (Burnside Theorem) Let G be a ﬁnite group and S ∈ Sylp(G).

Suppose NG(S) = CG(S) then there ∃ N: N / G such that G = S · N and S ∩ N=1.

12 Theorem 2.2.4 (Thompson Transfer Theorem) Let G be a ﬁnite group of even order, S ∈ Syl2(G) and M < S such that [S:M]=2. Suppose t ∈ S - M is an involution and tG∩ S ⊂ S - M then t 6∈ G0 .

Theorem 2.2.5 (Lemma 16 of [10]) Let G be a ﬁnite group of even order, S ∈

G Syl2(G) and M < S such that [S:M]=2. Suppose t ∈ S - M such that t ∩ S ⊂ S-M and suppose that (t2i )G ∩ S ⊂ M for ∀ integer i > 0 then t 6∈ G0 .

Theorem 2.2.6 (see [9]) Let G be a ﬁnite group of even order, S ∈ Syl2(G) and

S ∼= A × B where A is a non-trivial cyclic 2-group and B is a 2-group with a cyclic subgroup of index 2 then one of the following holds:

(1) S is an elementary abelian 2-group of order 4 or 8.

(2) [G : G0 ] is even.

(3) The group G/O20 (G) has a non-trivial normal 2-subgroup.

Theorem 2.2.7 (Theorem A∗ of [6]) Let G be a group with a non-abelian Sylow

2-subgroup which is the direct product of two dihedral groups. If G has no normal subgroups of index 2 and O(G) = 1 , then:

0 ∼ ∼ (i) G = L1 × L2, where L1 = A7 or PSL(2, q1), q1 odd, q1 ≥ 5, and L2 =

A7,PSL(2, q2), q2 odd, q2 ≥ 5, or Z2n × Z2n for some n;

(ii) G/G0 is of odd order and is abelian of rank at most 2.

13 2.3 Main consequences

In this section we shall prove some general Propositions that will be used in the

following sections to show that a certain 2-group is not realizable. We begin with the

following Proposition.

Proposition 2.3.1 Let G be a ﬁnite group of even order and S ∈ Syl2(G). Any

2-element of G that normalizes Z(S) centralizes it.

[Proof] Since S ⊂ CG(Z(S)), NG(Z(S))/CG(Z(S)) is of odd order so the propo-

sition follows.

Proposition 2.3.2 (Lemma 19 of [10]) Let G be a ﬁnite group of even order and

S ∈ Syl2(G). If Φ(S) is cyclic and for ∀ a ∈ S, CS(a) is not an elementary abelian

? group ⇒ Ω1(Φ(S)) ⊂ Z (G).

Proposition 2.3.3 Let G be a ﬁnite group of even order and S ∈ Syl2(G). If

G NG(S)/CG(S) is a 2-group then if z ∼ z1 where z, z1 ∈ Z(S) then z = z1.

|CG(S)|·|S| [Proof] NG(S)/CG(S) is a 2-group then |NG(S)| |CG(S)·S| = . CG(S)/ |S∩CG(S)| r NG(S) ⇒ S ∩ CG(S) ∈ Syl2(CG(S)) ⇒ |S ∩ CG(S)| = 2 for some integer r such

r that |CG(S)| = 2 · k and k is an odd integer such that |NG(S)| = |S| · k. So

62r·k·|S| |CG(S) · S| = 62r = |S| · k = |N|. So NG(S) = CG(S) · S.

Now suppose z is conjugate to z1 ∈ Z(S) in G then z and z1 are conjugate in NG(S)

k·n n = S·CG(S) so for k ∈ S and n ∈ CG(S), z = z1 = (as k ∈ S and z1 ∈ Z(S)) = z1

= (as n ∈ CG(S)) = z1.

14 Proposition 2.3.4 Let G be a ﬁnite group of even order and S ∈ Syl2(G). If for

∗ some integer i, |I2i (S)| < 3 · |I2i (Z(S))| then either |Z (G)| is even or ∃ z1 6= z2 ∈

G I2i (Z(S)) : z1 ∼ z2.

[Proof] Notice ﬁrst that for ∀ t ∈ I2i (S − Z(S)) there ∃ an element t1 ∈

S S − Z(S) such that t ∼ t1. If not t would be in the center. Now suppose no two ele-

∗ ments in I2i (Z(S)) are conjugate. Then we shall show that |Z (G)| is even. Pick any

G ∗ z ∈ I2i (Z(S)). Suppose z ∩ S = {z} then by Theorem 2.2.2 |Z (G)| is even. So now any z is conjugate to an element in I2i (S − Z(S)). But each element in I2i (S − Z(S)) is conjugate to at least one other element in I2i (S − Z(S)). So it follows that for any z ∈ I2i (Z(S)) there are at least 2 elements in I2i (S − Z(S)) conjugate to z. However

G |I2i (S)| < 3 · |I2i (Z(S))| so ∃ z ∈ I2i (Z(S)) such that z ∩ S = {z} ⇒ by Theorem

2.2.2 |Z∗(G)| is even.

Corollary 2.3.5 If NG(S)/CG(S) is a 2-group and for some integer i, |I2i (S)| <

∗ 3 · |I2i (Z(S))| then |Z (S)| is even.

∗ [Proof] By Proposition 2.3.4 either |Z (G)| is even or ∃z1 6= z2 ∈ I2i (Z(S)) such

G ∗ that z1 ∼ z2. Then by Proposition 2.3.3 |Z (G)| is even.

Proposition 2.3.6 Let G be a ﬁnite group of even order and S ∈ Syl2(G). If for some integer i, the number of conjugacy classes of elements of order 2i outside of Z(S)

∗ G is less than |I2i (Z(S))| then either |Z (G)| is even or ∃ z1 6= z2 ∈ I2i (Z(S)) : z1 ∼ z2.

15 [Proof] Suppose no two elements in I2i (Z(S)) are conjugate. Then we shall

∗ show that |Z (G)| is even. Pick any z ∈ I2i (Z(S)). Suppose z is conjugate to an

i element in I2i (S − Z(S)). However the number of central elements of order 2 is more than the number of conjugacy classes of elements outside of center. So at least two central elements of order 2i are fused in G as they are fused to elements in the same conjugacy class of S. So zG ∩ S = {z} then by Theorem 2.2.2 |Z∗(G)| is even.

Corollary 2.3.7 If NG(S)/CG(S) is a 2-group and for some integer i, the number

i of conjugacy classes of elements of order 2 outside of Z(S) is less than |I2i (Z(S))| then |Z∗(S)| is even.

∗ [Proof] By Proposition 2.3.6 either |Z (G)| is even or ∃z1 6= z2 ∈ Z(S) such

G ∗ that z1 ∼ z2. Then by Proposition 2.3.3 |Z (G)| is even.

Proposition 2.3.8 (Corollary 2 of [10]) Let G be a ﬁnite group of even order and

S ∈ Syl2(G). If S has a maximal subgroup M such that the order of every element in

M is less than the order of every element in S - M then [G : G0 ] is even.

Deﬁnition 2.3.9 A non-abelian 2-group S is called a generalized dihedral group if S contains an abelian subgroup M of index 2 such that the order of every element of S

- M is 2.

Deﬁnition 2.3.10 A 2-group S is called a DE-group if S is a direct product of a dihedral group with an elementary abelian group.

16 Proposition 2.3.11 (Lemma 17 of [10]) Let G be a ﬁnite group of even order and

0 S ∈ Syl2(G). If S is a generalized dihedral group then [G:G ] is even or S is a DE- group.

Proposition 2.3.12 (Lemma 18 of [10]) Let G be a ﬁnite group of even order and

S ∈ Syl2(G). If S contains a maximal subgroup M which is elementary abelian then

[G:G0 ] is even or S is a DE-group.

Deﬁnition 2.3.13 A 2-group S is called extra-special if |[S,S]| = |Z(S)| = |Φ(S)| =

Proposition 2.3.14 (Corollary 4 of [10]) Let G be a ﬁnite group of even order

∗ and S ∈ Syl2(G). If S is an extra-special 2-group then either Z(S) ⊂ Z (G) or ∼ S = D8.

Deﬁnition 2.3.15 (R. Brauer see [1]) Let G be a ﬁnite group of even order and S be one of the Sylow 2-subgroups of G. Let x ∈ S and K = S ∩ xG. An element y ∈ K is called an extreme element if |CS(y)| ≥ |CS(z)| for every z ∈ K.

Proposition 2.3.16 (Lemma 14 of [10]) If y is an extreme element of S, then

CS(y) is a Sylow 2-subgroup of CG(y). Furthermore, for every element z ∈ K there

θ exists an isomorphism θ of CS(z) into CS(y) such that z = y.

Proposition 2.3.17 (Theorem 8 of [4]) Let S be a Sylow 2-subgroup of a ﬁnite group G. Suppose S has the form Q × S0, where Q is a generalized quaternion group

n+1 of order 2 and where every involution in S0 lies in the n-th term of the upper

∗ central series, Zn(S0). Then |Z (G)| is even.

17 CHAPTER 3

DETERMINING REALIZABLE 2-GROUPS

3.1 2-groups of order ≤ 8

Although the proof for groups of order < 8 is trivial we present it here for completeness.

The following corollary of Burnside Theorem will be used to show that certain abelian 2-groups are not realizable.

Corollary 3.1.1 Let S be an abelian 2-group with an automorphism group Aut(S)

that is a 2-group. Then S is not realizable.

i [Proof] Aut(S) is a 2-group then |NG(S)/CG(S)| 2 . But as S is abelian S

⊂ CG(S) and S ∈ Syl2(NG(S)) so |NG(S)/CG(S)| is odd. Then it follows that

NG(S) = CG(S) and by Theorem 2.2.3 S is not realizable.

There is one group of order 2. It is not realizable.

∼ ∼ (1). S1 = [2, 1] = Z2 is not realizable by Corollary 3.1.1 as S is abelian and Aut(S)

is a 2-group.

There are 2 groups of order 4. One of them is realizable.

18 ∼ ∼ (1). S = S1 = [4, 2] = Z2 × Z2 ∈ Syl2(G) where G is isomorphic to one of the

2 3 following L2(4),L2(q), q ≡ 2 ± 1(mod 2 ) simple groups so S is a realizable group of

2 3 type L2(4),L2(q), q ≡ 2 ± 1(mod 2 ).

∼ ∼ (2). S2 = [4, 1] = Z4 is not realizable by Corollary 3.1.1 as S is abelian and Aut(S)

is a 2-group.

There are 5 groups of order 8. Two of them are realizable.

∼ ∼ (1). S = S1 = [8, 5] = Z2 × Z2 × Z2 ∈ Syl2(G) where G is isomorphic to one

2m+1 of the following J1,L2(8),R(3 ) simple groups so S is a realizable group of type

2m+1 J1,L2(8),R(3 ).

∼ ∼ ∼ ∼ (2). S2 = [8, 2] = Z4 × Z2, S3 = [8, 1] = Z8 are not realizable by Corollary 3.1.1 as S is abelian and Aut(S) is a 2-group.

∼ ∼ (3). S = S4 = [8, 3] = D8. S ∈ Syl2(G), where G is isomorphic to one of the

3 4 following A6,A7,L3(2),L2(q), q ≡ 2 ± 1(mod 2 ) simple groups. So S is a realizable

3 4 group of type A6,A7,L3(2),L2(q), q ≡ 2 ± 1(mod 2 ).

∼ ∼ G (4). S = S5 = [8, 4] = Q8. S has the only involution z so z ∩ S = {z} and by

Theorem 2.2.2 S is not realizable.

3.2 2-groups of order 16

Theorem 3.2.1 Let Si be a 2-group of order 16 numbered i on M. Hall, Jr. and J.K.

Senior list of groups of order 2n, n=4.(see [8]) There are 3 realizable groups of order ∼ ∼ ∼ ∼ ∼ ∼ 16. They are S1 = E16 = [16, 14], S12 = D16 = [16, 7] and S13 = SD16 = [16, 8].

19 [Proof] There are 14 groups of order 16. We shall consider every group S of order 16 and prove that either it can not be realizable or if it is realizable we shall present the group G for which S ∈ Syl2(G). That way we shall directly determine which groups on the list are not realizable without using classiﬁcation of ﬁnite simple groups.

∼ ∼ ∼ (1). S = S1 = [16, 14] = Z2 × Z2 × Z2 × Z2 ∈ Syl2(G) where G = L2(16) is a simple group so S is a realizable group of type L2(16).

∼ ∼ (2). S = S2 = [16, 10] = Z4 × Z2 × Z2. Let z be an element in S which is a square.

N (S) Suppose z ∼G y ∈ S then by Theorem 2.2.1, z G∼ y ∈ S. So a square goes into a square under a cojugation, because a conjugation is by an element in NG(S). But z is a unique square in S so it is ﬁxed. Then by Theorem 2.2.2 S is not realizable.

∼ ∼ ∼ ∼ (3). S = S3 = [16, 2] = Z4 × Z4. Then S = A × B where A = Z4 non-trivial cyclic ∼ group and B = Z4 is a 2-group with a cyclic subgroup Z2 of index 2. So by Theorem

0 2.2.6 as S is of order 16 either [G : G ] is even or G/O20 (G) has a non-trivial normal 2-subgroup so S is not realizable.

∼ ∼ (4). S = S4 = [16, 5], S5 = [16, 1] are not realizable by Corollary 3.1.1 as S is abelian and Aut(S) is a 2-group.

∼ ∼ ∼ (5). S = S6 = [16, 11]. S contains a maximal subgroup M = D8. Z(S) = Z2 × Z2 ∼ so S = Z2 × D8. ∼ ∼ ∼ S = S7 = [16, 12]. S contains a maximal subgroup M = Q8. Z(S) = Z2 × Z2 so S ∼ = Z2 × Q8.

Then by Theorem 2.2.6 S is not realizable.

20 ∼ ∼ 0 ∼ (6). S = S8 = [16, 11] = Z4 ∗ Q8. Y2 = S/S = Z2 × Z2 × Z2. S is a 2-group 0 ∼ so Φ(S) ⊃ S = Z2. But Φ(S) is smallest s.t. S/Φ(S) is elementary abelian ⇒ ∼ 0 ∼ ∼ Φ(S) = S = Z2 is cyclic. Also Z(S)= Z4 contains an element of order 4 so for

∀a ∈ S, CS(a) is not elementary abelian. Then by Proposition 2.3.2 it follows that S is not realizable.

∼ ∼ (7). S = S9 = [16, 3]. Z(S) = Z2 × Z2.

S has 7 involutions altogether and 3 involutions in the center so by Proposition 2.3.4

∗ either |Z (G)| is even and then by Theorem 2.2.2 S is not realizable or ∃ z, z1 ∈ Z(S)

G such that z ∼ z1. Aut(S) is a 2-group so NG(S)/CG(S) is also a 2-group so by

G Proposition 2.3.3 z = z1. Then as z ∩ S = {z} by Theorem 2.2.2 S is not realizable.

∼ ∼ (8). S = S10 = [16, 4]. Z(S) = Z2 × Z2. S has 3 involutions.

So all involutions of S are in the center so either zG∩ S = z and then by Theorem

G 2.2.2 S is not realizable or ∃ z, z1 ∈ Z(S) such that z ∼ z1. Aut(S) is a 2-group so

G NG(S)/CG(S) is also a 2-group so by Proposition 2.3.3 z = z1. Then as z ∩ S = {z} by Theorem 2.2.2 S is not realizable.

∼ (9). S = S11 = [16, 6]. Aut(S) is a 2-group so NG(S)/CG(S) is also a 2-group.

|I22 (S)| = 4, |I22 (Z(S))| = 2 so |I22 (S)| < 3 · |I22 (Z(S))| then by Corollary 2.3.5

|Z∗(S)| is even. So S is not realizable.

∼ ∼ (10). S = S12 = [16, 7] = D16. S ∈ Syl2(G), where G is isomorphic to one of the

4 5 following L2(q), q ≡ 2 ± 1(mod 2 ) simple groups. So S is a realizable group of type

4 5 L2(q), q ≡ 2 + 1(mod 2 ).

21 ∼ ∼ (11). S = S13 = [16, 8] = SD16. S ∈ Syl2(G), where G is isomorphic to one of

3 3 the following M11,L3(q), q ≡ 3(mod 2 ),U3(q), q ≡ 5(mod 2 ) simple groups. So S is

3 3 a realizable group of type M11,L3(q), q ≡ 3(mod 2 ),U3(q), q ≡ 5(mod 2 ).

∼ ∼ G (12). S = S14 = [16, 9] = Q16. S has the only involution z so z ∩ S = {z} and by

Theorem 2.2.2 S is not realizable.

3.3 2-groups of order 32

Theorem 3.3.1 Let Si be a 2-group of order 32 numbered i on M. Hall, Jr. and J.K.

Senior list of groups of order 2n, n=5.(see [8]) There are 4 realizable groups of order ∼ ∼ ∼ ∼ ∼ ∼ 32. They are S1 = E32 = [32, 51], S31 = Z4 o Z2 = [32, 11], S49 = D32 = [32, 18] and ∼ ∼ S50 = SD32 = [32, 19].

[Proof] There are 51 groups of order 32. We shall consider every group S of order 32 and prove that either it can not be realizable or if it is realizable we shall present the group G for which S ∈ Syl2(G). That way we shall directly determine which groups on the list are not realizable without using classiﬁcation of ﬁnite simple groups.

∼ ∼ ∼ (1). S = S1 = [32, 51] = Z2 × Z2 × Z2 × Z2 × Z2 ∈ Syl2(G) where G = L2(32) is a simple group so S is a realizable group of type L2(32).

∼ ∼ ∼ ∼ (2). S = S2 = [32, 45] = Z4 × Z2 × Z2 × Z2, S = S4 = [32, 36] = Z8 × Z2 × Z2.

Let z be an involution in S which is a square. Suppose z ∼G y ∈ S then by Theorem

N (S) 2.2.1, z G∼ y ∈ S. So a square goes into a square under a cojugation, because a

22 conjugation is by an element in NG(S). But z is a unique square in S so it is ﬁxed.

Then by Theorem 2.2.2 S is not realizable.

∼ ∼ (3). S = S3 = [32, 21] = Z4 × Z4 × Z2. Let M be a maximal subgroup of S

isomorphic to Z4 × Z4. All involutions in M are squares of elements of S. Let z be an

N (S) involution in S - M. Suppose z ∼G y ∈ M then by Theorem 2.2.1, z G∼ y ∈ M. So a

square goes into a square under a cojugation, because a conjugation is by an element

in NG(S). However z is not a square in S and any involution y ∈ M is a square so

we get a contradiction. Then by Theorem 2.2.4 S is not realizable.

∼ ∼ ∼ (4). S = S5 = [32, 3],S6 = [32, 16],S7 = [32, 1]. are not realizable by Corollary

3.1.1 as S is abelian and Aut(S) is a 2-group.

∼ ∼ ∼ (5). S = S8 = [32, 46] = Z2 × Z2 × D8 = D4 × D8 is a direct product of 2 dihedral

groups so by Theorem 2.2.7 it can not be realizable .

∼ ∼ ∗ (6). S = S9 = [32, 47] = Z2 × Z2 × Q8. So by Theorem 2.3.17 |Z (G)| is even so

S is not realizable.

∼ 0 ∼ (7). S = S10 = [32, 48].Y2 = S/S = Z2 × Z2 × Z2 × Z2. S is a 2-group so 0 ∼ Φ(S) ⊃ S = Z2. But Φ(S) is smallest such that S/Φ(S) is elementary abelian ⇒ ∼ 0 ∼ ∼ Φ(S) = S = Z2 is cyclic. Also Z(S) = Z4 × Z2 contains an element of order 4 so for

∀a ∈ S, CS(a) is not elementary abelian. ∼ 0 ∼ S = S17 = [32, 38]. Y2 = S/S = Z4 ×Z2 ×Z2. But Φ(S) is smallest such that S/Φ(S) is elementary abelian so Φ(S) is of order 4. It can’t be Z2 × Z2 as we know that S has ∼ elements of order 8 so it must be Z4 so cyclic. Also Z(S)= Z8 contains an element of order 4 so for ∀a ∈ S, CS(a) is not elementary abelian.

23 ∼ 0 ∼ 0 ∼ S = S26 = [32, 42]. Y2 = S/S = Z2 × Z2 × Z2. S is a 2-group so Φ(S) ⊃ S = Z4. But ∼ 0 ∼ Φ(S) is smallest such that S/Φ(S) is elementary abelian ⇒ Φ(S) = S = Z4 is cyclic. ∼ Also Z(S)= Z4 contains an element of order 4 so for ∀a ∈ S, CS(a) is not elementary abelian.

Then by Proposition 2.3.2 it follows that S is not realizable.

∼ ∼ (8). S = S11 = [32, 22]. Z(S) = Z2 × Z2 × Z2. S has 15 involutions altogether and

7 involutions in the center so 15 = |I21 (S)| < 3·|I21 (Z(S))| = 21. Aut(S) is a 2-group

∗ so NG(S)/CG(S) is also a 2-group and by Corollary 2.3.4 |Z (S)| is even so S is not

realizable.

∼ ∼ (9). S = S12 = [32, 23]. Z(S) = Z2 × Z2 × Z2. S has 7 involutions. ∼ ∼ ∼ S = S19 = [32, 4],S21 = [32, 12]. Z(S) = Z4 × Z2. S has 3 involutions. ∼ ∼ ∼ ∼ ∼ S = S28 = [32, 10],S29 = [32, 14],S30 = [32, 13],S35 = [32, 35],S40 = [32, 32]. Z(S) ∼ = Z2 × Z2. S has 3 involutions.

All involutions of S are in the center so |I21 (S)| = |I21 (Z(S))| < 3·|I21 (Z(S))|. Aut(S)

∗ is a 2-group so NG(S)/CG(S) is also a 2-group and by Corollary 2.3.4 |Z (S)| is even

so S is not realizable.

∼ ∼ ∼ ∼ (10). S = S13 = [32, 37],S16 = [32, 24],S20 = [32, 5]. Z(S) = Z4 × Z2. ∼ ∼ S = S37 = [32, 29]. Z(S) = Z2 × Z2.

S has 7 involutions altogether and 3 involutions in the center so 7 = |I21 (S)| <

3 · |I21 (Z(S))| = 9. Aut(S) is a 2-group so NG(S)/CG(S) is also a 2-group so by

Corollary 2.3.4 |Z∗(S)| is even so S is not realizable.

∼ ∼ ∼ (11). S = S14 = [32, 25]. S contains a maximal subgroup M = Z2 ×D8. Z(S) = Z4 ∼ so S = Z4 × D8.

24 ∼ ∼ ∼ S = S15 = [32, 26]. S contains a maximal subgroup M = Z2 × Q8. Z(S) = Z4 so S ∼ = Z4 × Q8. ∼ ∼ ∼ S = S23 = [32, 39]. S contains a maximal subgroup M = D16. Z(S) = Z2 × Z2 so ∃ ∼ an involution z ∈ Z(S) not in D16 which means that S = Z2 × D16. ∼ ∼ ∼ S = S24 = [32, 40]. S contains a maximal subgroup M = SD16. Z(S) = Z2 × Z2 so ∃ ∼ an involution z ∈ Z(S) not in SD16 which means that S = Z2 × SD16. ∼ ∼ ∼ S = S25 = [32, 41]. S contains a maximal subgroup M = Q16. Z(S) = Z2 × Z2 so ∃ ∼ an involution z ∈ Z(S) not in Q16 which means that S = Z2 × Q16.

Then by Theorem 2.2.6 S is not realizable.

∼ ∼ (12). S = S18 = [32, 2]. S has 7 involutions and Z(S) = Z2 × Z2 × Z2 so all

7 involutions of S are in the center. Notice that |Aut(S)| = 2 · 3 so |NG(S)/S · CG(S)|

= 1 or 3.

If |NG(S)/S · CG(S)| = 1 then NG(S) = S · CG(S). In this case for ∀z ∈ Z(S) and

g g ∀g ∈ G if z ∈ S then z = z. Indeed if z is fused with another involution z1 ∈ S then

z1 has to be in the center as all involutions of S are central. So by Theorem 2.2.1 the

conjugation can be done by an element g in NG(S). Then g = k · n where k ∈ S and

k·n n G n ∈ CG(S) and we have z = z = z. So z ∩ S = {z} and by Theorem 2.2.2 S is

not realizable.

Now suppose that |NG(S)/S · CG(S)| = 3. Similarly if z is fused with another invo-

lution it can be done by an element g in NG(S) by Theorem 2.2.1. If g is a 2-element

and it normalizes S it also normalizes Z(S) then by Proposition 2.3.1 g centralizes

Z(S). If g is of order 3 as there are 7 involutions altogether at least one involution

z ∈ Z(S) is ﬁxed by g. Hence zG∩ S = {z} and then by Theorem 2.2.2 S is not

realizable.

25 ∼ ∼ (13). S = S22 = [32, 17],S32 = [32, 15]. Aut(S) is a 2-group so NG(S)/CG(S) is also a 2-group. |I22 (S)| = 4, |I22 (Z(S))| = 2 so |I22 (S)| < 3 · |I22 (Z(S))| then by

Corollary 2.3.5 |Z∗(G)| is even.

∼ ∼ (14). S = S27 = [32, 9]. S has 11 involutions and Z(S) = Z2 × Z2 so there are 3 involutions in the center and 8 outside involutions. Let a ∈ S be an outside ∼ involution. Then CS(a) ⊃ Z(S) × < a > = Z2 × Z2 × Z2 so |CS(a)| ≥ 8. CS(a) is not of order 16 because none of the maximal subgroups of S contains a subgroup

Z2 × Z2 × Z2 with a in the center. So |CS(a)| = 8.

Now because |CS(a)| = 8 there are 2 conjugacy classes of outside involutions. Aut(S) is a 2-group and the number of central involutions is greater than the number of conjugacy classes of outside involutions. So by Corollary 2.3.7 |Z∗(G)| is even so S is not realizable.

∼ ∼ (15). S = S31 = [32, 11] = (Z4) o Z2. S ∈ Syl2(G), where G is isomorphic to one

2 3 2 3 of the following L3(q), q ≡ 2 + 1(mod 2 ),U3(q), q ≡ 2 − 1(mod 2 ) simple groups.

2 3 2 So S is a realizable group of type L3(q), q ≡ 2 + 1(mod 2 ),U3(q), q ≡ 2 − 1(mod

23).

∼ (16). S = S33 = [32, 27]. S contains an elementary abelian subgroup of order ∼ ∼ 16 and S is not a DE-group. Indeed S 6= Z2× Z2 × D8 because Z(S) = Z2 × Z2 ∼ and S 6= Z2 × D16 because S doesn’t contain a maximal subgroup D16. Then by

Proposition 2.3.12 it follows that [G:G0 ] is even so S is not realizable.

∼ (17). S = S34 = [32, 34]. S is non-abelian and S contains a maximal abelian ∼ subgroup M = Z4 × Z4 such that all the elements in S - M are involutions. Indeed

S has 12 elements of order 4 and so does M. Also S is not a DE-group because

26 ∼ ∼ ∼ S 6= Z2 × Z2 × D8 as Z(S) = Z2 × Z2 and S 6= Z2 × D16 because S doesn’t contain a

0 maximal subgroup D16. Then by Proposition 2.3.11 it follows that [G:G ] is even so

S is not realizable.

∼ ∼ ∼ ∼ (18). S = S36 = [32, 28],S38 = [32, 30],S39 = [32, 31],S41 = [32, 33]. S contains a ∼ 0 maximal abelian subgroup M such that Z(S) = Ω1(M) = Z2 × Z2. Suppose [G:G ] is odd then by Theorem 2.2.4 every involution in S is conjugate to an involution in M. All involutions in M are in Z(S), so every involution in S is conjugate to a

G central involution. In particular α5 ∼ α ∈ Z(S). So CG(α5) has a Sylow 2-subgroup isomorphic to S. ∼ CS(α5) =< α1, α2, α5 > ⊂ CG(α5) so ∃ a Sylow 2-subgroup of CG(α5) which contains

CS(α5). Let us denote such subgroup S1. Then Z(S1) ⊂ < α1, α2, α5 >. Indeed

CS(α5) is an elementary abelian subgroup of S1 of order 8 and S1 doesn’t have an elementary abelian subgroup of order 16. Therefore Z(S1) = < α5, α >, α ∈<

α1, α2, α1α2 >. Now because the only non-trivial commutators are [α3, α5] = α1,

[α4, α5] = α2 either α3, α4 or α3 · α4 normalizes Z(S1). But neither α3, α4 nor α3 · α4

ﬁxes α5 so we have a 2-element that normalizes Z(S1) but doesn’t centralize it. By

Proposition 2.3.1 we get a contradiction so [G:G0 ] is even so S is not realizable.

∼ ∼ ∼ ∼ (19). S = S42 = [32, 49],S43 = [32, 50]. [S,S] = Z(S) = Z2. Also S/Z(S) is an ∼ elementary abelian so Φ(S) = Z2 and S is an extra-special 2-group of order 32. Then by Proposition 2.3.14 Z(S) ⊂ Z∗(G) so S is not realizable.

∼ (20). S = S44 = [32, 43]. Let us use the following power-commutator presentation of S which is diﬀerent from the one used in [8]:

27 2 2 2 2 2 a1, a2, a3, a1 = a2 = a3 = a5 = 1, a4 = a5, S ∼= . a4, a5 [a1, a3] = [a1, a4] = [a2, a4] = a5, [a1, a2] = a4a5

S S S S S S has 5 conjugacy classes of involutions in S: (a1) ,(a2) ,(a2a3) ,(a3) and (a5) ∼ ∼ (where a5 is a central involution and Z(S) = Z2 =< a5 >) with the following cen- ∼ ∼ tralizers of involutions for each class: CS(a1) = Z2 × Z4, CS(a2) = Z2 × Z2 × Z2, ∼ ∼ ∼ CS(a2a3) = Z2 × Z2 × Z2, CS(a3) = Z2 × D8 and CS(a5) = S. There are 7 max- ∼ ∼ ∼ ∼ ∼ imal subgroups of S: M1 = D16 = [16, 7], M2 = [16, 13], M3 = D16 = [16, 7], ∼ ∼ ∼ ∼ ∼ ∼ ∼ M4 = Z2 ×D8 = [16, 11], M5 = SD16 = [16, 8], M6 = [16, 6] and M7 = SD16 = [16, 8].

Also we shall need to know what conjugacy classes of involutions are contained in each maximal subgroup:

S S S S S S S S S (a1) , (a2) , (a5) ∈ M1,(a1) , (a3) , (a5) ∈ M2,(a2a3) , (a5) , (a1) ∈ M3,

S S S S S S S S S S (a2) , (a2a3) , (a3) , (a5) ∈ M4,(a2) , (a5) ∈ M5,(a3) , (a5) ∈ M6,(a2a3) , (a5) ∈

M7.

G Suppose now a3 ∼ a5 then by Proposition 2.3.16 there exists an isomorphism θ from ∼ ∼ CS(a3) = Z2 × D8 into CS(a5) = S that maps a3 into a5. But CS(a3) has a unique

G square a5 so it will be mapped into a5 under θ. We get a contradiction. So a3 6∼ a5.

G G We can similarly show that a1 6∼ a5. Indeed suppose a1 ∼ a5 then by Proposition ∼ ∼ 2.3.16 there exists an isomorphism θ from CS(a1) = Z2 × Z4 into CS(a5) = S that maps a1 into a5. But CS(a3) has a unique square a5 so it will be mapped into a5 under θ. We get a contradiction.

Notice that a5 has to be fused with another involution as otherwise by Glauberman

∗ S Z -Theorem 2.2.2 S can not be realizable. Also as a1 6∈ M6 and M6 contains (a3) and

S (a5) then a1 has to be fused with either a3 or a5 as otherwise by Thompson Transfer

G Theorem 2.2.4 S can not be realizable. So it follows that a1 ∼ a3. As a2, a2a3 6∈ M6

28 S S and M6 contains (a3) and (a5) then each a2 and a2a3 has to be fused with either a3

or a5 as otherwise by Thompson Transfer Theorem 2.2.4 S can not be realizable. So

there are 4 possible cases. Now we just consider the fusion patterns and show that

for each case S can not be realizable group.

G G ∗ Case 1. Suppose a2 ∼ a2a3 ∼ a3. Then a5 is isolated and by Glauberman Z -Theorem

2.2.2 S can not be realizable.

G G G G Case 2. Suppose a2 ∼ a2a3 ∼ a5 or a2 ∼ a5, a2a3 ∼ a3. Then as a3 6∈ M5 and M5

S S contains (a2) and (a5) then a3 has to be fused with either a2 or a5 as otherwise by

Thompson Transfer Theorem 2.2.4 S can not be realizable. We get a contradiction.

G G S Case 3. Suppose a2 ∼ a3 and a2a3 ∼ a5. Then as a3 6∈ M7 and M7 contains (a2a3)

S and (a5) then a3 has to be fused with either a2 or a5 as otherwise by Thompson

Transfer Theorem 2.2.4 S can not be realizable. We get a contradiction.

So S can not be realizable.

∼ ∼ (21). S = S45 = [32, 44]. S/ < z >= Z2 × D8 where z is central involution. So 0 ∼ ∼ 0 ∼ S/S = Z2 × Z2 × Z2. Then it follows that Φ(S) = S = Z4. So Φ(S) is cyclic. S does not have subgroups isomorphic to Z2 × Z2 × Z2 and Z2 × Z2 × Z2 × Z2 so if CS(a) ∼ is an elementary abelian then it has to be isomorphic to Z2 × Z2. But then S = D2n

or SD2n . That is a contradiction so CS(a) is not an elementary abelian for ∀a ∈ S.

? Now by Proposition 2.3.2 it follows that Ω1(Φ(S)) ⊂ Z (G) so S is not realizable.

∼ ∼ ∼ ∼ (22). S = S46 = [32, 6]. Consider M = CS(α2) =< α2, α3, α4 > = Z2 × D8.

M is a maximal subgroup of S and it is unique subgroup isomorphic to Z2 × D8. ∼ Notice that < α1 > is the commutator subgroup of M = Z2 × D8 and M/ < α1 >

∼ ∼ 2 = Z2 × D8/ < α1 >= Z2 × Z2 × Z2 so y = α1 for ∀y ∈ M. Since M contains all 11

29 involutions of S if x ∈ S − M is not fused with an element y ∈ M then by Theorem

2.2.4 [G : G0 ] is even. Suppose [G : G0 ] is odd then x ∼G y ∈ M for ∀x ∈ S −M. Hence

2 G 2 2 x ∼ y = α1. However α2α3 = (α5 · α4) is a square of an element α5 · α4 ∈ S − M,

G so it follows that α2α3 ∼ α1. Then by Proposition 2.3.16 there ∃ an isomorphism ∼ Θ 0 ∼ Θ from M = CS(α2α3) into S, such that (α2α3) = α1. But M =< α1 > for any

Θ maximal subgroup of S so α1 = α1. We get a contradiction.

∼ ∼ (23). S = S47 = [32, 7],S48 = [32, 8]. S contains a maximal subgroup M such that any element in M is of order at most 4 and any element outside of M is of order 8.

Then by Proposition 2.3.8 [G : G0 ] is even so S is not realizable.

∼ ∼ (24). S = S49 = [32, 18] = D32. S ∈ Syl2(G), where G is isomorphic to one of the

5 6 following L2(q), q ≡ 2 ± 1(mod 2 ) simple groups. So S is a realizable group of type

5 6 L2(q), q ≡ 2 + 1(mod 2 ).

∼ ∼ (25). S = S50 = [32, 19] = SD32. S ∈ Syl2(G), where G is isomorphic to one of

3 4 3 4 the following L3(q), q ≡ 2 − 1(mod 2 ),U3(q), q ≡ 2 + 1(mod 2 ) simple groups. So

3 4 3 4 S is a realizable group of type L3(q), q ≡ 2 − 1(mod 2 ),U3(q), q ≡ 2 + 1(mod 2 ).

∼ ∼ G (26). S = S51 = [32, 20] = Q32. S has the only involution z so z ∩ S = {z} and by Theorem 2.2.2 S is not realizable.

3.4 2-groups of order 64

There are 267 groups of order 64. It is not practical to eliminate all 2-groups of order 64 that can not be realizable by hand. We shall attempt to use a computer to check some internal properties of a group to determine if it can be realizable.

30 The following is the algorithm that is implemented using GAP 4 commands to elimi-

nate all 2-groups of a given order that can not be realizable.

INPUT: n, where 2n is the order of 2-groups.

OUTPUT: R, an array consisting of pairs of integers, ﬁrst integer is a 2-group ID

number in GAP and second integer from 1 to 11 will label the reason that a 2-group can not be realizable. If it is 0 we can not eliminate the group. for i from 1 to (number of 2-groups of order 2n) do;

S:=i-th 2-group of order 2n;

R[i][1]:=i; R[i][2]:=0; ***** S may be realizable *****

if ( S ∼= A × B, where A is non-trivial cyclic and B has a subgroup of index 2 that is

cyclic ) then

R[i][1]:=i; R[i][2]:=1; ***** S is not realizable, reason 1. *****

end if;

if ( ( S is abelian ) and ( S has a unique involution ) ) then

R[i][1]:=i; R[i][2]:=2; ***** S is not realizable, reason 2. *****

end if;

if ( ( S is not abelian ) and ( S is not isomorphic to A × D2n , where A is elementary

abelian ) and ( there is a maximal subgroup M of S which is abelian and every element

of S - M is an involution) ) then

R[i][1]:=i; R[i][2]:=3; ***** S is not realizable, reason 3. *****

end if;

if ( ( S is not abelian ) and ( S is not isomorphic to A × D2n , where A is elementary abelian ) and ( there is a maximal subgroup M isomorphic to elementary abelian

31 group) ) then

R[i][1]:=i; R[i][2]:=4; ***** S is not realizable, reason 4. ***** end if; if ( there is a maximal subgroup M of S such that the order of every element of M is less than the order of every element of S - M ) then

R[i][1]:=i; R[i][2]:=5; ***** S is not realizable, reason 5. *****

end if;

if ( S is extra-special ) then

R[i][1]:=i; R[i][2]:=6; ***** S is not realizable, reason 6. *****

end if;

if ( Frattini subgroup of S is cyclic ) then

if ( Center of S is NOT elementary abelian ) then

R[i][1]:=i; R[i][2]:=7; ***** S is not realizable, reason 7. *****

else

if ( CS(a) is NOT elementary abelian for any a ∈ S) then

R[i][1]:=i; R[i][2]:=7; ***** S is not realizable, reason 7. *****

end if;

if ( Aut(S) is a 2-group ) then

if ( S is abelian ) then

R[i][1]:=i; R[i][2]:=8; ***** S is not realizable, reason 8. *****

else ***** S is NOT abelian, continue *****

if ( 3 * ( number of elements of order 2i in the center ) >

32 ( number of elements of S of order 2i )) then

R[i][1]:=i; R[i][2]:=9; ***** S is not realizable, reason 9. *****

else

if ( number of involutions in the center > number of conjugacy

classes of involutions outside of the center ) then

R[i][1]:=i; R[i][2]:=10; ***** S is not realizable, reason 10. *****

end if;

end if; end if; if ( ( S has 7 involutions, all of them in Z(S) ) and ( order of Aut(S) is 2j ·3)) then

R[i][1]:=i; R[i][2]:=11; ***** S is not realizable, reason 11. *****

end if;

end do;

To show that the algorithm is correct we need to show that each time a 2-group

is eliminated it can not be realizable. We summarize all the cases in the following

theorem.

Theorem 3.4.1 After the algorithm terminates only groups that are labeled with 0

can be realizable.

[Proof] (1). If a 2-group S is labeled with 1 then the reason it was eliminated

is that S ∼= A × B where A is a non-trivial cyclic group, B has a cyclic subgroup of

33 index 2 and the order of S is larger than 8 then by Theorem 2.2.6 [G : G0 ] is even or

G/O20 (G) has a non-trivial normal 2-subgroup. So S is not realizable.

(2). If a 2-group S is labeled with 2 then the reason it was eliminated is that if

G z is a unique square of S and if z1 ∼ z then by Proposition 2.3.16 there exists an ∼ ∼ isomorphism θ from CS(z1) = S into CS(z) = S that maps z1 into z. But CS(z1) has a unique square z, so z will be mapped into z under θ. We get a contradiction with

G ∗ z1 ∼ z. That means z is an isolated involution and by Glauberman Z -Theorem 2.2.2

S can not be realizable.

(3). If a 2-group S is labeled with 3 then the reason it was eliminated is that S ∼ is non-abelian and S 6= A × D2n , where A is elementary abelian and also S has a maximal subgroup M which is abelian and every element of S - M is an involution.

Then by Proposition 2.3.11 [G:G0 ] is even. So S is not realizable.

(4). If a 2-group S is labeled with 4 then the reason it was eliminated is that ∼ S is non-abelian and S 6= A × D2n , where A is elementary abelian and also S has a maximal subgroup M which is isomorphic to elementary abelian group. Then by

Proposition 2.3.12 [G:G0 ] is even. So S is not realizable.

(5). If a 2-group S is labeled with 5 then the reason it was eliminated is that S has a maximal subgroup M such that the order of every element of M is less than the order of every element of S - M, then by Proposition 2.3.8 [G : G0 ] is even. So S is

34 not realizable.

(6). If a 2-group S is labeled with 6 then the reason it was eliminated is that S is an extra-special group and order of S is larger than 8, then by Proposition 2.3.14

Z(S) ⊂ Z∗(G). So S is not realizable.

(7). If a 2-group S is labeled with 7 then the reason it was eliminated is that

Frattini subgroup of S is cyclic and for ∀ a ∈ S, CS(a) is not an elementary abelian

? group so by Proposition 2.3.2 Ω1(Φ(S)) ⊂ Z (G). So S is not realizable.

(8). If a 2-group S is labeled with 8 then the reason it was eliminated is that S is an abelian group with automorphism group that is a 2-group then by Corollary 3.1.1

S is not realizable.

(9). If a 2-group S is labeled with 9 then the reason it was eliminated is that

∗ Aut(S) is a 2-group and |I2i (S)| < 3·|I2i (Z(S))| so by Corollary 2.3.5 |Z (S)| is even.

So S is not realizable.

(10). If a 2-group S is labeled with 10 then the reason it was eliminated is that

Aut(S) is a 2-group and there are more central involutions of S than there are conjugacy classes of non-central involutions. So by Corollary 2.3.7 |Z∗(S)| is even and S is not realizable.

35 (11). If a 2-group S is labeled with 11 then the following holds. Notice that

j |Aut(S)| = 2 · 3 so |NG(S)/S ·CG(S)| = 1 or 3. If NG(S) = S · CG(S) any element z ∈ Z(S) is ﬁxed under a conjugation by an element g ∈ NG(S). Indeed for g = k · n

k·n n G where k ∈ S and n ∈ CG(S) we have z = z = z so z ∩ S = {z} and then by

Theorem 2.2.2 S is not realizable.

Now suppose that |NG(S)/S · CG(S)| = 3. So NG(S) induces an automorphism σ ∼ of order 3 on Z(S). However S has 7 involutions and Z(S) = Z2 × Z2 × Z2 so all involutions of S are in the center. It follows then that at least one involution z ∈ Z(S)

is ﬁxed by σ. Then by Proposition 2.3.1 z is ﬁxed by any automorphism of S so zG∩

S = {z} and then by Theorem 2.2.2 S is not realizable.

36 CHAPTER 4

PROPERTIES OF REALIZABLE GROUPS OF ORDER 26

In this chapter we list the properties of Realizable Groups of order 26. We use programs implemented in GAP4 to make most computations.

Notation 4.0.2 We use the following notation:

S = [2n, k] - k-th group of order 2n in the Small Group Library in GAP 4.

S is of type G if S ∈ Syl2(G).

In power-commutator presentation we use commutator formula [a, b] = a−1b−1ab for a, b ∈ G.

There are 267 groups of order 26, 8 of them are realizable groups.

They are: [64, 267], [64, 52], [64, 53], [64, 82], [64, 245], [64, 138], [64, 242] and [64, 134].

4.1 Realizable group of type L2(64)

S is a Sylow-2 subgroup of simple groups L2(64).

S is an elementary abelian group of order 64. ∼ S = [64, 267] = Z2 × Z2 × Z2 × Z2 × Z2 × Z2.

37 2 2 2 2 2 2 f1, f2, f3, f1 = f2 = f3 = f4 = f5 = f6 = 1, S ∼= f4, f5, f6 [fi, fj] = 1 ∀ i, j

∼ Z(S) = Z2.

S/Z(S) ∼= Id is not a realizable group.

There are 63 maximal subgroups of S isomorphic to [32, 51]. ∼ All of them are realizable groups of lower order isomorphic to:[32, 51] = Z2 × Z2 ×

Z2 × Z2 × Z2 of type L2(32).

S/Φ(S) ∼= Id. ∼ Maximal quotient group of S is isomorphic to [32,51] = Z2 × Z2 × Z2 × Z2 × Z2.

There are 63 maximal abelian subgroup of S.

Abelian Group A Z2 × Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 1

NG(A)/CG(A) is isomorphic to Z2

They are also maximal elementary abelian subgroups of S.

S has no normal extra-special subgroups.

S has 63 involutions in 63 conjugacy classes.

All involutions are fused in G.

6 7 4.2 Realizable group of type L2(q), q ≡ 2 ± 1(mod 2 )

6 7 S is a Sylow-2 subgroup of simple groups L2(q), q ≡ 2 ± 1(mod 2 ). ∼ S = [64, 52] = D64.

38 ∼ 2 32 2 S = f1, f2 f1 = f2 = 1, [f1, f2] = f2

∼ Z(S) = Z2.

∼ ∼ 5 6 S/Z(S) = [32, 18] = D32 is a realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ). ∼ Note that [32,18] = D32 occurs among the maximal subgroups of S.

There are 3 maximal subgroups of S: [32, 18], [32, 18], [32, 1].

Two of them are realizable groups of lower order:

∼ 5 6 Both are [32, 18] = D32 of type L2(q), q ≡ 2 ± 1(mod 2 ). ∼ S/Φ(S) = Z2 × Z2. ∼ Maximal quotient group of S is isomorphic to [32,18]= D32.

There is 1 maximal abelian subgroup of S.

Abelian Group A Z32 Number of groups in each conjugacy class 1 ∼ If G = L2(191),L2(193) then

NG(A)/CG(A) is isomorphic to Z2

There are 16 elementary abelian subgroups of S in 2 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 Number of groups in each conjugacy class 8+8 ∼ If G = L2(191),L2(193) then

NG(A)/CG(A) is isomorphic to S3

39 There is no extra fusion that occurs in G for the maximal elementary abelian subgroups.

S has no normal extra-special subgroups.

S has 33 involutions in 3 conjugacy classes.

S S 16 S Conjugacy class (f1) (f1f2) (f2 ) Number of involutions 16 16 1

All involutions are fused in G.

4 5 4.3 Realizable group of type L3(q), q ≡ 2 −1(mod 2 ),U3(q), q ≡ 24 + 1(mod 25)

4 5 S is a Sylow-2 subgroup of simple groups L3(q), q ≡ 2 −1(mod 2 ) and U3(q), q ≡

24 + 1(mod 25). ∼ S = [64, 53] = SD64.

∼ 2 32 24−2 S = f1, f2 f1 = f2 = 1, [f2, f1] = f2

∼ Z(S) = Z2.

∼ ∼ 5 6 S/Z(S) = [32, 18] = D32 is a realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ). ∼ Note that [32,18] = D32 occurs among the maximal subgroups of S.

There are 3 maximal subgroups of S: [32, 20], [32, 18], [32, 1].

One of them is realizable group of lower order:

∼ 5 6 [32, 18] = D32 of type L2(q), q ≡ 2 ± 1(mod 2 ).

40 ∼ S/Φ(S) = Z2 × Z2. ∼ Maximal quotient group of S is isomorphic to [32,18] = D32.

There is 1 maximal abelian subgroup of S.

Abelian Group A Z32 Number of groups in each conjugacy class 1 ∼ If G = L3(47),U3(17) then

NG(A)/CG(A) is isomorphic to Z2

There are 8 maximal elementary abelian subgroups of S in 1 conjugacy class.

Elementary Abelian Group A Z2 × Z2 Number of groups in each conjugacy class 8 ∼ If G = L3(47),U3(17) then

NG(A)/CG(A) is isomorphic to S3

There is no extra fusion that occurs in G for the maximal elementary abelian subgroups.

S has no normal extra-special subgroups.

S has 17 involutions in 2 conjugacy classes.

S 32 S Conjugacy class (f1) (f2 ) Number of involutions 16 1

All involutions are fused in G.

41 4.4 Realizable group of type Sz(8)

S is a Sylow-2 subgroup of simple group Sz(8).

S = [64, 82].

2 2 2 2 2 2 f1, f2, f3, f = f4f5f6, f = f4f5, f = f4, f = f = f = 1, S ∼= 1 2 3 4 5 6 f4, f5, f6 [f2, f1] = f4, [f3, f1] = f5, [f3, f2] = f6

∼ Z(S) = Z2 × Z2 × Z2. ∼ ∼ S/Z(S) = [8, 5] = Z2 × Z2 × Z2 is a realizable group of type L2(8).

There are 7 maximal subgroups of S isomorphic to [32, 2]. None of them are realizable

groups. ∼ S/Φ(S) = Z2 × Z2 × Z2.

The following are maximal quotient groups of S: ∼ If z ∈ f4, f5, f6, f4f5, f4f6, f5f6, f4f5f6 then S/ < z >= [32,33].

There are 7 maximal abelian subgroups of S in 7 conjugacy classes.

Abelian Group A Z2 × Z2 × Z4 Number of groups in each conjugacy class 1+1+1+1+1+1+1

NG(A)/CG(A) is isomorphic to Z2 × Z2

There is no extra fusion that occurs in G for maximal abelian subgroups of S.

There is 1 elementary abelian subgroup of S isomorphic to Z2 × Z2 × Z2.

Elementary Abelian Group A Z2 × Z2 × Z2 Number of groups in each conjugacy class 1

NG(A)/CG(A) is isomorphic to Z7

42 There are no normal extra-special subgroups of S.

S has 7 central involutions:f4, f5, f6, f4f5, f4f6, f5f6, f4f5f6

All involutions are fused in G.

4.5 Realizable group of type U3(4)

S is a Sylow-2 subgroup of simple group U3(4).

S = [64, 245].

2 2 2 2 2 2 f1, f2, f3, f = f4f5, f = f = f4f5f6, f = f = f6, f = 1, [f2, f1] = f4f5, S ∼= 1 2 3 4 5 6 f4, f5, f6 [f3, f1] = [f4, f2] = [f5, f2] = f4f5f6, [f3, f2] = [f4, f3] = [f5, f3] = f6

∼ Z(S) = Z2 × Z2. ∼ S/Z(S) = [16, 14] is a realizable group of type L2(16).

There are 15 maximal subgroups of S, all of them are isomorphic to [32, 32]. None of ∼ them are realizable groups. S/Φ(S) = Z2 × Z2 × Z2 × Z2.

The following are maximal quotient groups of S: ∼ If z = f5, f6, f5f6 then S/ < z >= [32,50].

There are 5 maximal abelian subgroups of S of order 16 isomorphic to Z4 × Z4 in 5 conjugacy classes.

Abelian Group A Z4 × Z4 Number of groups in each conjugacy class 1+1+1+1+1

NG(A)/CG(A) is isomorphic to A4

43 All maximal abelian subgroups isomorphic to Z4 × Z4 are fused in G.

There is 1 maximal elementary abelian subgroup of S of order 4 isomorphic to Z2 ×Z2.

Elementary Abelian Group A Z2 × Z2 Number of groups in each conjugacy class 1

NG(A)/CG(A) is isomorphic to Z3

There is no extra fusion that occurs in G for the maximal abelian subgroups.

There are no normal extra-special subgroup of S.

S has 3 central involutions in 3 conjugacy classes.

2 S 2 S S Conjugacy class (f1 ) (f2 ) (f6) Number of involutions 1 1 1

All involutions are fused in G.

2 4.6 Realizable group of type A8,A9,L4(2),U4(2),S4(q), q ≡ 2 ± 1(mod 23)

2 S is a Sylow-2 subgroup of simple groups A8,A9,L4(2),U4(2),S4(q), q ≡ 2 ±1(mod

23).

S = [64, 138].

2 2 2 2 2 2 ∼ f1 = f2 = f3 = f4 = f5 = f6 = 1, [f2, f1] = f4, S = f1, f2, f3, f4, f5, f6 [f3, f1] = f5, [f5, f2] = [f4, f3] = f6

∼ Z(S) = Z2.

S/Z(S) ∼= [32, 27] is not a realizable group.

44 There are 7 maximal subgroups of S, 3 of them are isomorphic to [32, 27], 3 groups are isomorphic to [32, 6] and the other group is: [32, 49]. None of them are realizable

groups. ∼ S/Φ(S) = Z2 × Z2 × Z2.

The following are maximal quotient groups of S: ∼ If z = f6 then S/ < z >= [32,27].

There is 1 maximal abelian subgroup of S of order 16.

Abelian Group A Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 1 ∼ If G = A8,A9,L4(2)

then NG(A)/CG(A) is isomorphic to A5

It is also maximal elementary abelian subgroup of S.

There is 1 maximal normal extra-special subgroup of S: [32, 49].

Extra-special Group E [32, 49] Number of groups in each conjugacy class 1 ∼ If G = A8,A9,L4(2)

then NG(E)/CG(E) is isomorphic to [96, 227] ∼ 2 3 If G = U4(2),S4(q), q ≡ 2 ± 1(mod 2 )

then NG(E)/CG(E) is isomorphic to [288, 1025]

S has 27 involutions in 9 conjugacy classes.

S S S S S S S Conjugacy class (f1) (f2) (f3) (f4) (f5) (f6) (f1f6) Number of involutions 4 4 4 2 2 1 4

45 ∼ If G = A8,A9,L4(2) then there are 2 conjugacy classes of involutions in G with the following number of involutions in each class: 2+4+4, 1+2+2+4+4+4.

∼ 2 3 If G = U4(2),S4(q), q ≡ 2 ±1(mod 2 ) then there are 2 conjugacy classes of involutions in G with the following number of involutions in each class: 1+4, 2+2+2+4+4+4+4.

4.7 Realizable group of type L3(4)

S is a Sylow-2 subgroup of simple groups L3(4).

S = [64, 242].

2 2 2 2 2 2 f1, f2, f3, f = f = f = f = f = f = 1, [f3, f1] = f4, S ∼= 1 2 3 4 5 6 f4, f5, f6 [f6, f1] = [f3, f2] = f4f5, [f6, f2] = f5

∼ Z(S) = Z2 × Z2. ∼ ∼ S/Z(S) = Z2 × Z2 × Z2 × Z2 = [16, 14] is a realizable group of type L2(16).

There are 15 maximal subgroups of S, 9 of them are isomorphic to [32, 31], 6 groups are isomorphic to [32, 27]. None of them are realizable groups. ∼ S/Φ(S) = Z2 × Z2 × Z2 × Z2.

The following are maximal quotients group of S: ∼ If z ∈ f5, f4, f4f5 then S/ < z >= [32,49].

There are 5 maximal abelian subgroups of S of order 16 in 5 conjugacy classes.

Abelian Group A Z2 × Z2 × Z2 × Z2 Z4 × Z4 Number of groups in each conjugacy class 1+1 1+1+1

NG(A)/CG(A) is isomorphic to A5 A4

46 2 of them are elementary abelian subgroups of S of order 16 in 2 conjugacy classes.

None of maximal elementary abelian subgroups isomorphic to Z2 × Z2 × Z2 × Z2 are fused in G. Also none of maximal abelian subgroups isomorphic to Z4 × Z4 are fused in G.

There are no normal extra-special subgroups of S.

S has 27 involutions in 9 conjugacy classes.

S S S S S S Conjugacy class (f4) (f5) (f4f5) (f1) (f2) (f1f2) Number of involutions 1 1 1 4 4 4

S S S Conjugacy class (f5f6) (f4f5f3) (f4f3f6) Number of involutions 4 4 4

All involutions are fused in G.

2 3 4.8 Realizable group of type M12, D4(q), q ≡ 2 ± 1(mod 2 ), 2 3 G2(q), q ≡ 2 ± 1(mod 2 )

2 3 S is a Sylow-2 subgroup of simple groups M12, D4(q), q ≡ 2 ± 1(mod 2 ),

2 3 G2(q), q ≡ 2 ± 1(mod 2 ).

S = [64, 134].1

2 2 2 2 2 2 f1, f2, f3, f1 = f2, f2 = f3 = f6 = 1, f4 = f5 = f6, [f3, f1] = f2, S ∼= f4, f5, f6 [f4, f1] = [f4, f3] = f5, [f4, f2] = [f5, f3] = [f5, f4] = f6

∼ Z(S) = Z2.

1 ∼ 2 3 For group G = D4(q), q ≡ 2 ± 1(mod 2 ) the fusion of elements of S in G is not computed.

47 S/Z(S) ∼= [32, 27] is not a realizable group.

There are 7 maximal subgroups of S: [32, 43], [32, 43], [32, 11], [32, 11], [32, 34], [32, 49] and [32, 7].

2 Two groups isomorphic to [32, 11] are realizable groups of type L3(q), q ≡ 2 ± 1(mod

3 2 3 2 ),U3(q), q ≡ 2 ± 1(mod 2 ). ∼ S/Φ(S) = Z2 × Z2 × Z2.

The following is maximal quotient group of S: ∼ If z = f1 then S/ < z >= [32,27].

There is 1 maximal abelian subgroup of S of order 16.

Abelian Group A Z4 × Z4 Number of groups in each conjugacy class 1 ∼ If G = M12

then NG(A)/CG(A) is isomorphic to S3 × Z2

There are 8 elementary abelian subgroups of S of order 8 in 5 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 × Z2 Number of groups in each conjugacy class 1+1+2+2+2 ∼ If G = M12

then NG(A)/CG(A) is isomorphic to S4 + S4 + S4 + S4 + A4

There is no extra fusion that occurs in G for maximal abelian and elementary abelian subgroups of S.

There is 1 maximal normal extra-special subgroup of S: [32, 49].

48 Extra-special Group E [32, 49] Number of groups in each conjugacy class 1 ∼ If G = M12 then

NG(E)/CG(E) is isomorphic to [96, 227] ∼ If G = G2(3),G2(5) then

NG(E)/CG(E) is isomorphic to [288, 1026]

S has 27 involutions in 7 conjugacy classes.

S S S S Conjugacy class (f3) (f6) (f2) (f3f1) Number of involutions 8 1 2 4

S −1 S S Conjugacy class (f4f2) (f3f6f5 f1) (f4f3f1) Number of involutions 4 4 4

∼ There are 2 conjugacy classes of involutions in G, if G = M12.

The ﬁrst one consists of 4 conjugacy classes of involutions of S fused in G:

S S S S Conjugacy class (f3) ∪ (f6) ∪ (f2) ∪ (f3f1) Number of involutions 8+1+2+4

The second one consists of 3 conjugacy classes of involutions of S fused in G:

S −1 S S Conjugacy class (f4f2) ∪ (f3f6f5 f1) ∪ (f4f3f1) Number of involutions 4+4+4

∼ All involutions are fused in G for G = G2(3),G2(5).

49 CHAPTER 5

REALIZABLE GROUPS OF ORDER 27

In this chapter we list the properties of Realizable Groups of order 27.

Notation 5.0.1 We use the following notation:

S = [2n, k] - k-th group of order 2n in the Small Group Library in GAP 4.

S is of type G if S ∈ Syl2(G).

In power-commutator presentation we use commutator formula [a, b] = a−1b−1ab for a, b ∈ G.

There are 2328 groups of order 27, 7 of them are realizable groups.

They are: [128, 2328], [128, 161], [128, 162], [128, 67], [128, 928], [128, 931] and [128, 934].

5.1 Realizable group of type L2(128)

S is a Sylow-2 subgroup of simple groups L2(128).

S is an elementary abelian group of order 128. ∼ S = [128, 2328] = Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2.

50 2 2 2 2 2 2 2 f1, f2, f3, f4, f = f = f = f = f = f = f = 1, S ∼= 1 2 3 4 5 6 7 f5, f6, f7 [fi, fj] = 1 ∀ i, j

∼ Z(S) = Z2.

S/Z(S) ∼= Id is not a realizable group.

There are 127 maximal subgroups of S isomorphic to [64, 267]. ∼ All of them are realizable groups of lower order isomorphic to:[64, 267] = Z2 × Z2 ×

Z2 × Z2 × Z2 × Z2 of type L2(64).

S/Φ(S) ∼= Id. ∼ Maximal quotient group of S is isomorphic to [64,267] = Z2 × Z2 × Z2 × Z2 × Z2 × Z2.

There are 127 maximal abelian subgroup of S.

Abelian Group A Z2 × Z2 × Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 1

NG(A)/CG(A) is isomorphic to Z2

They are also maximal elementary abelian subgroups of S.

S has no normal extra-special subgroups.

S has 127 involutions in 127 conjugacy classes.

All involutions are fused in G.

7 8 5.2 Realizable group of type L2(q), q ≡ 2 ± 1(mod 2 )

7 8 S is a Sylow-2 subgroup of simple groups L2(q), q ≡ 2 ± 1(mod 2 ). ∼ S = [128, 161] = D128.

51 ∼ 2 64 2 S = f1, f2 f1 = f2 = 1, [f1, f2] = f2

∼ Z(S) = Z2.

∼ ∼ 6 7 S/Z(S) = [64, 52] = D64 is a realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ). ∼ Note that [64,52] = D64 occurs among the maximal subgroups of S.

There are 3 maximal subgroups of S: [64, 52], [64, 52], [64, 1].

Two of them are realizable groups of lower order:

∼ 6 7 Both are [64, 52] = D64 of type L2(q), q ≡ 2 ± 1(mod 2 ). ∼ S/Φ(S) = Z2 × Z2. ∼ Maximal quotient group of S is isomorphic to [64, 52] = D64.

There is 1 maximal abelian subgroup of S.

Abelian Group A Z64 Number of groups in each conjugacy class 1 ∼ If G = L2(129),L2(127) then

NG(A)/CG(A) is isomorphic to Z2

There are 32 elementary abelian subgroups of S in 2 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 Number of groups in each conjugacy class 16+16 ∼ If G = L2(129),L2(127) then

NG(A)/CG(A) is isomorphic to: S3

There is no extra fusion that occurs in G for the maximal elementary abelian subgroups.

S has no normal extra-special subgroups.

52 S has 65 involutions in 3 conjugacy classes.

S S 32 S Conjugacy class (f1) (f1f2) (f2 ) Number of involutions 32 32 1

All involutions are fused in G.

5 6 5.3 Realizable group of type L3(q), q ≡ 2 − 1(mod 2 ) and 5 6 U3(q), q ≡ 2 + 1(mod 2 )

5 6 S is a Sylow-2 subgroup of simple groups L3(q), q ≡ 2 −1(mod 2 ) and U3(q), q ≡

25 + 1(mod 26). ∼ S = [128, 162] = SD128.

∼ 2 64 25−2 S = f1, f2 f1 = f2 = 1, [f2, f1] = f2

∼ Z(S) = Z2.

∼ ∼ 6 7 S/Z(S) = [64, 52] = D64 is a realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ). ∼ Note that [64,52] = D64 occurs among the maximal subgroups of S.

There are 3 maximal subgroups of S: [64, 54], [64, 52], [64, 1].

One of them is realizable group of lower order:

∼ 6 7 [64, 52] = D64 of type L2(q), q ≡ 2 ± 1(mod 2 ). ∼ S/Φ(S) = Z2 × Z2. ∼ Maximal quotient group of S is isomorphic to [64, 52] = D64.

There is 1 maximal abelian subgroup of S.

53 Abelian Group A Z64 Number of groups in each conjugacy class 1 ∼ If G = L3(31),U3(33) then

NG(A)/CG(A) is isomorphic to Z2

There are 16 elementary abelian subgroups of S in 1 conjugacy class.

Elementary Abelian Group A Z2 × Z2 Number of groups in each conjugacy class 16 ∼ If G = L3(31),U3(33) then

NG(A)/CG(A) is isomorphic to S3

S has no normal extra-special subgroups.

S has 33 involutions in 2 conjugacy classes.

S 32 S Conjugacy class (f1) (f2 ) Number of involutions 32 1

All involutions are fused in G.

3 4 5.4 Realizable group of type L3(q), q ≡ 2 +1(mod 2 ),U3(q), q ≡ 23 − 1(mod 24)

3 4 S is a Sylow-2 subgroup of simple groups L3(q), q ≡ 2 + 1(mod 2 ),U3(q), q ≡

23 − 1(mod 24). ∼ S = [128, 67] = Z8 o Z2.

54 2 2 2 2 2 2 * f1, f2, f3, f1 = f2f3f6, f2 = f7 = 1, f3 = f4 = f6, f5 = f6 = f7, + ∼ S = f4, f5, f6, [f2, f1] = [f3, f1] = [f4, f1] = f3f4f5f6, [f3, f2] = f5,

f7 [f5, f2] = [f5, f3] = [f5, f4] = f7, [f4, f3] = f5f7

∼ Z(S) = Z8.

∼ ∼ 4 5 S/Z(S) = [16, 7] = D16 is a realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ). ∼ There are 3 maximal subgroups of S: [64, 45], [64, 145] and [64, 2] = Z8 × Z8. None of them are realizable groups. ∼ S/Φ(S) = Z2 × Z2.

The following is maximal quotient group of S: ∼ If z = f7 then S/ < z >= [64,6].

There is 1 maximal abelian subgroup of S .

Abelian Group A Z8 × Z8 Number of groups in each conjugacy class 1 ∼ If G = L3(9),U3(7) then

NG(A)/CG(A) is isomorphic to S3

There are 5 elementary abelian subgroups of S in 2 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 Number of groups in each conjugacy class 1+4 ∼ If G = L3(9),U3(7) then

NG(A)/CG(A) is isomorphic to S3

∼ For G = L3(9),U3(7) maximal elementary abelian subgroups isomorphic to Z2 × Z2 are fused in G.

There is 1 maximal normal extra-special subgroup of S: [32, 42].

55 Extra-special Group E [32, 42] Number of groups in each conjugacy class 1 ∼ If G = L3(9),U3(7) then

NG(E)/CG(E) is isomorphic to [16, 7]

S has 11 involutions in 3 conjugacy classes.

S −1 −1 S S Conjugacy class (f7) (f5 f6 ) (f2) Number of involutions 1 2 8

∼ All involutions are fused in G for G = L3(9),U3(7).

5.5 Realizable group of type A10

S is a Sylow-2 subgroup of simple groups A10. ∼ S = [128, 928] = D8 o Z2.

2 2 2 2 2 2 2 * f1 = f2 = f3 = f4 = f5 = f6 = f7 = 1, [f1, f3] = + ∼ f1, f2, f3, f4, S = [f2, f3] = f2f1, [f4, f6] = [f5, f6] = f4f5, [f1, f7] = [f4, f7] = . f5, f6, f7 f4f1, [f3, f7] = [f6, f7] = f3f6, [f2, f7] = [f5, f7] = f2f5

∼ Z(S) = Z2. ∼ S/Z(S) = [64, 138] is a realizable group of type A8.

There are 7 maximal subgroups of S: [64, 138], [64, 138], [64, 32], [64, 32], [64, 134],

[64, 226] and [64, 34].

Three of them are realizable groups.

The group [64, 134] is of type M12 and two groups [64, 138] are of type A8.

56 ∼ S/Φ(S) = Z2 × Z2 × Z2.

The following is maximal quotient group of S: ∼ If z = f7 then S/ < z >= [64,138].

There are 9 maximal abelian subgroups of S of order 16 in 6 conjugacy classes.

Abelian Group A Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 1+1+2

NG(A)/CG(A) is isomorphic to S5+[72,40]+(S3 × S3)

Elementary Abelian Group A Z2 × Z2 × Z4 Z4 × Z4 Number of groups in each conjugacy class 2+2 1

NG(A)/CG(A) is isomorphic to (S3 × Z2)+(S3 × Z2) D8

Four of them are elementary abelian subgroups of S of order 16 in 3 conjugacy classes.

There is no extra fusion that occurs in G for the maximal abelian subgroups.

There is 1 maximal normal extra-special subgroup of S: [32, 49].

Extra-special Group E [32, 49] Number of groups in each conjugacy class 1

NG(E)/CG(E) is isomorphic to [192, 955]

S has 43 involutions in 10 conjugacy classes.

S S S S S S Conjugacy class (f1) (f2f1) (f3) (f4f1) (f4f2f1) (f4f3) Number of involutions 4 2 4 4 4 8

S S S S Conjugacy class (f4f2f1f5) (f4f3f5) (f3f6) (f7) Number of involutions 1 4 4 8

57 There are 2 conjugacy classes of involutions in G.

The ﬁrst one consists of 4 conjugacy classes of involutions of S fused in G:

S S S S Conjugacy class (f1) ∪ (f2f1) ∪ (f3) ∪ (f4f1) Number of involutions 4+2+4+4

The second one consists of 6 conjugacy classes of involutions of S fused in G:

S S S Conjugacy class (f4f2f1) ∪ (f4f3) ∪ (f4f2f1f5) ∪ S S S (f4f3f5) ∪ (f3f6) ∪ (f7) Number of involutions 4+8+1+4+4+8

5.6 Realizable group of type M22, M23, McL, L4(q), q ≡ 5(mod 3 3 2 ), U4(q), q ≡ 3(mod 2 )

3 S is a Sylow-2 subgroup of simple groups M22, M23, McL, L4(q), q ≡ 5(mod 2 )

3 and U4(q), q ≡ 3(mod 2 ).

S = [128, 931].

2 2 2 2 2 2 2 * f1 = f2 = f3 = f4 = f5 = f6 = f7 = 1, + ∼ f1, f2, f3, f4, S = [f1, f3] = f5, [f1, f6] = [f2, f3] = [f4, f5] = f7, f5, f6, f7 [f1, f2] = f4, [f3, f4] = f6, [f2, f5] = f6f7

∼ Z(S) = Z2. ∼ S/Z(S) = [64, 138] is a realizable group of type A8.

There are 7 maximal subgroups of S: [64, 138], [64, 138], [64, 32], [64, 32], [64, 136],

[64, 242] and [64, 35]. Three of them are realizable groups. [64, 242] is of type L3(4)

and two groups [64, 138] are of type A8. ∼ S/Φ(S) = Z2 × Z2 × Z2 × Z2.

58 The following is maximal quotient group of S: ∼ If z = f7 then S/ < z >= [64,138].

There are 5 maximal abelian subgroups of S of order 16 in 4 conjugacy classes.

Abelian Group A Z2 × Z2 × Z2 × Z2 Z4 × Z4 Number of groups in each conjugacy class 1+1 1+2 ∼ If G = M22 then

NG(A)/CG(A) is isomorphic to S5+A6 S4+A4 ∼ If G = M23 then [360,120]+

NG(A)/CG(A) is isomorphic to Group of order 2520 S4+S4 If G ∼= McL then Group of order 2520+

NG(A)/CG(A) is isomorphic to Group of order 2520 S4+S4 ∼ If G = L4(5),U4(3) then

NG(A)/CG(A) is isomorphic to A6+A6 S4+A4

2 of them are elementary abelian subgroups of S of order 16 in 2 conjugacy classes. ∼ For G = M22, L4(5) and U4(3) there is no extra fusion that occurs in G for maximal abelian and maximal elementary abelian subgroups for S. ∼ For G = M23, McL maximal abelian subgroups isomorphic to Z4 × Z4 are fused in G

and there is no extra fusion that occurs for maximal elementary abelian subgroups

of S.

There is 1 maximal normal extra-special subgroup of S: [32, 49].

59 Extra-special Group E [32, 49] Number of groups in each conjugacy class 1 ∼ If G = M22,M23

then NG(E)/CG(E) is isomorphic to [192, 955] ∼ If G = McL, L4(5), U4(3) then

NG(E)/CG(E) is isomorphic to [576, 8654]

S has 35 involutions in 7 conjugacy classes.

S S S S S S S Conjugacy class (f1) (f2) (f3) (f4) (f5) (f6) (f7) Number of involutions 8 8 8 4 4 2 1

All involutions are fused in G.

5.7 Realizable group of type J2, J3

2 S is a Sylow-2 subgroup of simple groups J2,J3.

S = [128, 934].

2 2 2 2 2 2 2 f1 = f4f5, f2 = f3f4f7, f3 = f4 = f7, f5 = f6 = f7 = 1, ∼ f1, f2, f3, f4, S = [f2, f1] = f3f5f6f7, [f3, f1] = f4f5f7, [f4, f1] = [f5, f1] = f4f5f6, f5, f6, f7 [f6, f1] = [f6, f2] = [f4, f3] = [f6, f3] = [f6, f4] = [f6, f5] = f7,

[f3, f2] = [f4, f2] = f3f4, [f5, f2] = f3f4f6f7

∼ Z(S) = Z2.

2 ∼ Although matrix and permutation representations of group G = J3 are known, none of them can be used in GAP to compute in reasonable time the fusion of elements of S in G. We compute ∼ the fusion in G = J2 only.

60 ∼ S/Z(S) = [64, 138] is a realizable group of type A8.

There are 7 maximal subgroups of S: [64, 135], [64, 135], [64, 135], [64, 36], [64, 36],

[64, 36] and [64, 242]. One of them is a realizable group: [64, 242] is of type L3(4). ∼ S/Φ(S) = Z2 × Z2 × Z2 × Z2.

The following is maximal quotient group of S: ∼ If z = f7 then S/ < z >= [64,138].

There are 5 maximal abelian subgroups of S of order 16 in 4 conjugacy classes.

Abelian Group A Z2 × Z2 × Z2 × Z2 Z4 × Z4 Number of groups in each conjugacy class 2 1+1+1 ∼ If G = J2 then

NG(A)/CG(A) is isomorphic to (A4 × Z3) S4

Two of them are elementary abelian subgroups of S of order 16 in 1 conjugacy class. ∼ All maximal abelian subgroups isomorphic to Z4 × Z4 are fused in G if G = J2.

There is 1 maximal normal extra-special subgroup of S: [32, 50].

Extra-special Group E [32, 50] Number of groups in each conjugacy class 1 ∼ If G = J2 then

NG(E)/CG(E) is isomorphic to [960, 11358]

S has 35 involutions in 6 conjugacy classes.

S S S −2 −1 S Conjugacy class (f5) (f6) (f7) (f5f6f1 f2 ) Number of involutions 8 2 1 8

61 −1 −1 S −1 −1 S Conjugacy class (f1 f2 ) (f1 f3 ) Number of involutions 8 8

∼ If G = J2 are 2 conjugacy classes of involutions in G.

The ﬁrst one consists of 3 conjugacy classes of involutions of S fused in G:

S S S Conjugacy class (f5) ∪ (f6) ∪ (f7) Number of involutions 8+2+1

The second one consists of 3 conjugacy classes of involutions of S fused in G:

−2 −1 S −1 −1 S −1 −1 S Conjugacy class (f5f6f1 f2 ) ∪ (f1 f2 ) ∪ (f1 f3 ) Number of involutions 8+8+8

62 CHAPTER 6

REALIZABLE GROUPS OF ORDER 28

In this chapter we list the properties of Realizable Groups of order 28.

Notation 6.0.1 We use the following notation:

S = [2n, k] - k-th group of order 2n in the Small Group Library in GAP 4.

S is of type G if S ∈ Syl2(G).

In power-commutator presentation we use commutator formula [a, b] = a−1b−1ab for a, b ∈ G.

There are 56092 groups of order 28, 7 of them are realizable groups.

They are: [256, 56092], [256, 8935], [256, 539], [256, 540], [256, 5298], [256, 6661] and

[256, 6665].

6.1 Realizable group of type L2(256)

S is a Sylow-2 subgroup of simple groups L2(256).

S is an elementary abelian group of order 128.

63 ∼ ∼ S = [256, 56092] = Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2.

2 2 2 2 2 2 ∼ f1, f2, f3, f4, f1 = f2 = f3 = f4 = f5 = f6 = S = 2 2 f5, f6, f7, f8 f7 = f8 = 1, [fi, fj] = 1 ∀ i, j

∼ Z(S) = Z2.

S/Z(S) ∼= Id is not a realizable group.

There are 255 maximal subgroups of S isomorphic to [128, 2328]. ∼ All of them are realizable groups of lower order isomorphic to:[128, 2328] = Z2 × Z2 ×

Z2 × Z2 × Z2 × Z2 × Z2 of type L2(128).

S/Φ(S) ∼= Id. ∼ Maximal quotient group of S is isomorphic to [128,2328] = Z2 × Z2 × Z2 × Z2 × Z2 ×

Z2 × Z2.

There are 255 maximal abelian subgroup of S.

Abelian Group A Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2 Number of groups in 1 each conjugacy class

NG(A)/CG(A) is isomorphic to Z2

They are also maximal elementary abelian subgroups of S.

S has no normal extra-special subgroups.

S has 255 involutions in 255 conjugacy classes.

All involutions are fused in G.

64 6.2 Realizable group of type S4(4)

S is a Sylow-2 subgroup of simple group S4(4).

S ∼= [256, 8935].

2 2 2 2 2 2 2 * f1, f2, f3, f1 = f2 = f3 = f5 = f6 = f7 = f8 = 1, + ∼ 2 S = f4, f5, f6, f4 = f5f7, [f8, f1] = f7, [f4, f1] = f3f5f6f7, .

f7, f8 [f4, f2] = f5f7, [f8, f2] = [f8, f4] = f6

∼ Z(S) = Z2 × Z2 × Z2 × Z2. ∼ ∼ S/Z(S) = [16, 14] = Z2 × Z2 × Z2 × Z2 is a realizable group of type L2(16).

There are 15 maximal subgroups of S:

Maximal subgroup [128, 170] [128, 2163] Number of groups 9 6

None of them are realizable groups. ∼ S/Φ(S) = Z2 × Z2 × Z2 × Z2.

The following are maximal quotient groups of S: ∼ If z ∈ {f5f3f7f6, f7f6, f5f7f6, f3f7, f5f7, f7, f6, f5f3f6, f3f6} then S/ < z >= [128,1135]. ∼ If z ∈ {f5, f3, f5f3, f3f7f6, f5f3f7, f5f6} then S/ < z >= [128,1411].

There are 2 maximal abelian subgroups of S in 2 conjugacy classes.

Both groups are maximal elementary abelian subgroups of S.

65 Abelian Group A Z2 × Z2 × Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 1+1 ∼ If G = S4(4) then

NG(A)/CG(A) is isomorphic to A5 × Z3

There are no normal extra-special subgroups of S.

S has 111 involutions in 39 conjugacy classes.

There are 15 conjugacy classes of central involutions:

S S S S S S S S S (f5) ,(f3) ,(f5f3) ,(f3f7f6) ,(f5f3f7) ,(f5f6) ,(f5f3f7f6) ,(f7f6) ,(f5f7f6) ,

S S S S S S (f3f7) ,(f5f7) (f7) ,(f6) ,(f5f3f6) ,(f3f6) .

There are 24 conjugacy classes of involutions with 4 involutions in each class:

S S 2 S −1 S S −1 S S (f5f7f1) ,(f5f2) ,(f5f2f1f4 ) ,(f7f2f4 ) ,(f7f8) ,(f2f4 f8) ,(f7f1) ,

S S S S S 2 S 2 S (f5f3f7f1) ,(f3f7f1) ,(f2) ,(f5f3f2) ,(f3f2) ,(f2f1f4 ) ,(f5f3f2f1f4 ) ,

2 S −1 S −1 S −1 S S S (f3f2f1f4 ) ,(f5f7f2f4 ) ,(f3f7f2f4 ) ,(f5f3f7f2f4 ) ,(f5f7f8) ,(f3f7f8) ,

S −1 S −1 S −1 S (f5f3f7f8) ,(f5f2f4 f8) ,(f3f2f4 f8) ,(f5f3f2f4 f8) . There are 3 conjugacy classes of involutions in G.

The ﬁrst one consists of 6 conjugacy classes of involutions of S fused in G:

S S S S Conjugacy class (f5) ∪ (f3) ∪ (f5f3) ∪ (f5f7f1) ∪ S 2 S (f5f2) ∪ (f5f2f1f4 ) Number of involutions 1+1+1+4+4+4

66 The second one consists of 6 conjugacy classes of involutions of S fused in G:

S S S −1 S Conjugacy class (f3f7f6) ∪ (f5f3f7) ∪ (f5f6) ∪ (f7f2f4 ) ∪ S S (f7f8) ∪ (f2f4f8) Number of involutions 1+1+1+4+4+4

The third one consists of 27 conjugacy classes of involutions of S fused in G:

S S S S S Conjugacy class (f5f3f7f6) ∪ (f7f6) ∪ (f5f7f6) ∪ (f3f7) ∪ (f5f7) ∪ S S S S S (f7) ∪ (f6) ∪ (f5f3f6) ∪ (f3f6) ∪ (f7f1) ∪ S S S S S (f5f3f7f1) ∪ (f3f7f1) ∪ (f2) ∪ (f5f3f2) ∪ (f3f2) ∪ 2 S 2 S 2 S −1 S (f2f1f4 ) ∪ (f5f3f2f1f4 ) ∪ (f3f2f1f4 ) ∪ (f5f7f2f4 ) ∪ −1 S −1 S S (f3f7f2f4 ) ∪ (f5f3f7f2f4 ) ∪ (f5f7f8) ∪ S S (f3f7f8) ∪ (f5f3f7f8)) ∪ −1 S −1 S −1 S (f5f2f4 f8) ∪ (f3f2f4 f8) ∪ (f5f3f2f4 f8) Number of involutions 1+1+1+1+1+1+1+1+1+4+4+4+4+4+4+ 4+4+4+4+4+4+4+4+4+4+4+4

8 9 6.3 Realizable group of type L2(q), q ≡ 2 ± 1(mod 2 )

8 9 S is a Sylow-2 subgroup of simple groups L2(q), q ≡ 2 ± 1(mod 2 ). ∼ ∼ S = [256, 539] = D256.

∼ 2 128 2 S = f1, f2 f1 = f2 = 1, [f1, f2] = f2 .

∼ Z(S) = Z2.

∼ ∼ 7 8 S/Z(S) = [128, 161] = D128 is a realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ). ∼ Note that [128,161] = D128 occurs among the maximal subgroups of S.

67 There are 3 maximal subgroups of S: [128, 161], [128, 161] and [128, 1].

Two of them are realizable groups of lower order:

∼ 7 8 Both are [128, 161] = D128 of type L2(q), q ≡ 2 ± 1(mod 2 ). ∼ S/Φ(S) = Z2 × Z2. ∼ Maximal quotient group of S is isomorphic to [128,161] = D128.

There is 1 maximal abelian subgroup of S.

Abelian Group A Z128 Number of groups in each conjugacy class 1 ∼ If G = L2(1279),L2(257) then

NG(A)/CG(A) is isomorphic to Z2

There are 64 elementary abelian subgroups of S in 2 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 Number of groups in each conjugacy class 32+32 ∼ If G = L2(1279),L2(257) then

NG(A)/CG(A) is isomorphic to S3+S3

There is no extra fusion that occurs in G for the maximal elementary abelian subgroups.

S has no normal extra-special subgroups.

S has 129 involutions in 3 conjugacy classes.

S S 64 S Conjugacy class (f1) (f1f2) (f2 ) Number of involutions 64 64 1

68 All involutions are fused in G.

6 7 6.4 Realizable group of type L3(q), q ≡ 2 −1(mod 2 ),U3(q), q ≡ 26 + 1(mod 27)

6 7 S is a Sylow-2 subgroup of simple groups L3(q), q ≡ 2 −1(mod 2 ) and U3(q), q ≡

26 + 1(mod 27). ∼ ∼ S = [256, 540] = SD256.

∼ 2 128 26−2 S = f1, f2 f1 = f2 = 1, [f2, f1] = f2 .

∼ Z(S) = Z2.

∼ ∼ 7 8 S/Z(S) = [128, 161] = D128 is a realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ). ∼ Note that [128,161] = D128 occurs among the maximal subgroups of S.

There are 3 maximal subgroups of S: [128, 161], [128, 163] and [128, 1]. ∼ One of them is realizable group of lower order: [128, 161] = D128 of type L3(q), q ≡

27 ± 1(mod 28). ∼ S/Φ(S) = Z2 × Z2. ∼ Maximal quotient group of S is isomorphic to [128,161] = D128.

There is 1 maximal abelian subgroup of S.

69 Abelian Group A Z128 Number of groups in each conjugacy class 1 ∼ If G = L3(191),U3(193) then

NG(A)/CG(A) is isomorphic to Z2

There are 32 elementary abelian subgroups of S in 1 conjugacy class.

Elementary Abelian Group A Z2 × Z2 Number of groups in each conjugacy class 32 ∼ If G = L3(191),U3(193) then

NG(A)/CG(A) is isomorphic to S3

There is no extra fusion that occurs in G for the maximal elementary abelian subgroups.

S has no normal extra-special subgroups.

S has 65 involutions in 2 conjugacy classes.

S 64 S Conjugacy class (f1) (f2 ) Number of involutions 64 1

All involutions are fused in G.

70 3 4 6.5 Realizable group of type G2(q), q ≡ 2 ±1(mod 2 ),D4(q), q ≡ 23 ± 1(mod 24)

3 4 S is a Sylow-2 subgroup of simple groups G2(q), q ≡ 2 ± 1(mod 2 ),D4(q), q ≡

23 ± 1(mod 24).

S ∼= [256, 5298].

2 2 2 2 2 2 f1, f2, f3, f1 = f2 = f3 = f6 = f8 = 1, f4 = f6, ∼ 2 2 S = f4, f5, f6, f5 = f7f8, f7 = f8, [f2, f1] = f4, [f3, f1] = f5,

f7, f8 [f4, f1] = [f4, f2] = f6, [f5, f1] = [f5, f2] = f7,

[f7, f1] = [f7, f2] = f8, [f4, f3] = [f5, f3] = f7, [f6, f3] = [f7, f3] = f8

∼ Z(S) = Z2.

S/Z(S) ∼= [128, 351] is not a realizable group.

Note that [128,351] does not occur among the maximal subgroups of S.

There are 7 maximal subgroups of S:

[128, 444], [128, 945], [128, 945], [128, 2024], [128, 89], [128, 67] and [128, 67].

Two of them are realizable groups of lower order:

3 4 Both are [128, 67] of type L3(q), q ≡ 2 + 1(mod 2 ). ∼ S/Φ(S) = Z2 × Z2 × Z2.

Maximal quotient group of S is isomorphic to [128,351].

There is 1 maximal abelian subgroup of S.

71 Abelian Group A Z8 × Z8 Number of groups in each conjugacy class 1 ∼ If G = G2(7) then

NG(A)/CG(A) is isomorphic to S3 × Z2

There are 32 elementary abelian subgroups of S in 5 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 × Z2 Number of groups in each conjugacy class 8+4+4+8+8 ∼ If G = G2(7) then

NG(A)/CG(A) is isomorphic to [168, 42]+[168, 42]+S4+S4+S4

There are no normal extra-special subgroups of S.

S has 83 involutions in 7 conjugacy classes.

S S S S S S S Conjugacy class (f1) (f2) (f3) (f6) (f8) (f2f3) (f2f5) Number of involutions 32 16 8 2 1 8 16

3 4 6.6 Realizable group of type S4(q), q ≡ 2 ± 1(mod 2 )

3 4 S is a Sylow-2 subgroup of simple groups S4(q), q ≡ 2 ± 1(mod 2 ).

S ∼= [256, 6661].

2 2 2 2 2 2 2 f2 = f3 = f4 = f5 = f6 = f7 = f8 = 1, ∼ f1, f2, f3, f4 2 S = f1 = f3f4f5f6f8, [f4, f1] = [f6, f1] = f4f6f7, f5, f6, f7, f8 [f3, f1] = [f5, f1] = f3f5f7f8, [f5, f2] = [f6, f2] = f5f6,

72 [f5, f3] = f7f8, [f6, f4] = f7, [f3, f2] = [f4, f2] = f3f4, [f7, f2] = f8

∼ Z(S) = Z2. ∼ S/Z(S) = [128, 928] is a realizable group of type A10.

Note that [128,928] occurs among the maximal subgroups of S.

There are 7 maximal subgroups of S:

[128, 928], [128, 928], [128, 351], [128, 2024], [128, 134], [128, 134] and [128, 140].

Two of them are realizable groups of lower order:

They are both isomorphic to [128, 928] of type A10. ∼ S/Φ(S) = Z2 × Z2 × Z2.

Maximal quotient group of S is isomorphic to [128,928].

There is 1 maximal abelian subgroup of S.

Abelian Group A Z4 × Z8 Number of groups in each conjugacy class 1 ∼ If G = S4(7),S4(9) then

NG(A)/CG(A) is isomorphic to D8

There are 4 elementary abelian subgroups of S in 2 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 2+2 ∼ If G = S4(7),S4(9) then

NG(A)/CG(A) is isomorphic to S5

73 ∼ There is no extra fusion that occurs in G = S4(7),S4(9) for the maximal elementary abelian subgroups of S.

There are 2 maximal normal extra-special subgroups E of S: [32, 49], [32, 49].

Extra-special Group E [32, 49] Number of groups in each conjugacy class 1+1 ∼ If G = S4(7),S4(9) then

NG(E)/CG(E) is isomorphic to [1152, 157849]

S has 67 involutions in 9 conjugacy classes.

S S S S −1 S Conjugacy class (f6) (f8) (f6f5) (f6f8) (f6f5f1 ) Number of involutions 8 1 8 8 16

S S S S Conjugacy class (f7) (f6f3) (f6f1f8f2f4) (f2f3f4) Number of involutions 2 8 8 8

There are 2 conjugacy classes of involutions in G.

The ﬁrst one consists of 2 conjugacy classes of involutions of S fused in G:

S S Conjugacy class (f6) ∪ (f8) Number of involutions 8+1

The second one consists of 7 conjugacy classes of involutions of S fused in G:

S S −1 S S Conjugacy class (f6f5) ∪ (f6f8) ∪ (f6f5f1 ) ∪ (f7) ∪ S S S (f6f3) ∪ (f6f1f8f2f4) ∪ (f2f3f4) Number of involutions 8+8+16+2+8+8+8

74 6.7 Realizable group of type Ly

∼ 3 S = Z2(Z2 o Z2 o Z2) is a Sylow-2 subgroup of simple group Ly.

S ∼= [256, 6665].

2 2 2 2 2 2 f1, f2, f3 f1 = f2 = f3 = f5 = f7 = f8 = 1, [f3, f1] = f5, ∼ 2 2 S = f4, f5, f6, f4 = f6 = f8, [f4, f1] = [f4, f2] = [f7, f2] =

f7, f8 [f6, f3] = [f7, f3] = [f6, f4] = [f6, f5] = f8, [f2, f1] = f4,

[f5, f2] = f6f7f8, [f6, f1] = f7, [f4, f3] = f6, [f5, f4] = f7f8

∼ Z(S) = Z2. ∼ S/Z(S) = [128, 928] is a realizable group of type A10.

Note that [128,928] does not occur among the maximal subgroups of S.

There are 7 maximal subgroups of S: [128, 934], [128, 931], [128, 387], [128, 2020],

[128, 136], [128, 135] and [128, 138].

Two of them are realizable groups of lower order: [128, 931] is of type M22, M23, McL,

3 3 L4(q), q ≡ 5(mod 2 ) and U4(q), q ≡ 3(mod 2 ) and [128, 934] is of type J2. ∼ S/Φ(S) = Z2 × Z2 × Z2.

Maximal quotient group of S is isomorphic to [128,928].

There are 19 maximal abelian subgroups of S in 7 conjugacy classes.

Abelian Group A Z4 × Z4 Z2 × Z8 Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 1+2 2+4+4+4 2

3Although matrix representations of group G =∼ Ly are known, none of them can be used in GAP to compute the fusion of elements of S in G. We used generators and relations of S given in [5] to ﬁnd the ID number of S in Small Groups library in GAP.

75 Two maximal abelian subgroups are maximal elementary abelian subgroups of S isomorphic to Z2 × Z2 × Z2 × Z2. They are conjugate in S.

There are 2 maximal normal extra-special subgroups of S: [32, 49] and [32, 50].

S has 59 involutions in 7 conjugacy classes.

S S S S S S S Conjugacy class (f1) (f2) (f3) (f5) (f7) (f8) (f2f3) Number of involutions 16 8 8 8 2 1 16

76 CHAPTER 7

REALIZABLE GROUPS OF ORDER 29

In this chapter we list the properties of Realizable Groups of order 29.

Notation 7.0.1 We use the following notation:

S = [2n, k] - k-th group of order 2n in the Small Group Library in GAP 4.

S is of type G if S ∈ Syl2(G).

In power-commutator presentation we use commutator formula [a, b] = a−1b−1ab for a, b ∈ G.

There are 10494213 groups of order 29, 12 of them are realizable groups.

They are: [512, 10494213], [512, 2042], [512, 2043], [512, 947], [512, 60809], [512, 60833],

[512, 7530110], [512, 406983], [512, n1], [512, n2], [512, 60329] and [512, 58362], where

7532393 ≤ n1, n2 ≤ 10481221.

7.1 Realizable group of type L2(512)

S is a Sylow-2 subgroup of simple groups L2(512).

S is an elementary abelian group of order 512.

77 ∼ ∼ S = [512, 10494213] = Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2.

2 2 2 2 2 * f1, f2, f3, f1 = f2 = f3 = f4 = f5 = + ∼ 2 2 2 2 S = f4, f5, f6, f6 = f7 = f8 = f9 = 1,

f7, f8, f9 [fi, fj] = 1 ∀ i, j

∼ Z(S) = Z2.

S/Z(S) ∼= Id is not a realizable group.

There are 511 maximal subgroups of S isomorphic to [256, 56092]. ∼ All of them are realizable groups of lower order isomorphic to:[256, 56092] = Z2 ×

Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2 of type L2(256).

S/Φ(S) ∼= Id. ∼ Maximal quotient group of S is isomorphic to [256,56092] = Z2 × Z2 × Z2 × Z2 × Z2 ×

Z2 × Z2 × Z2.

There are 511 maximal abelian subgroup of S.

Abelian Group A Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2 Number of groups in 1 each conjugacy class

NG(A)/CG(A) is isomorphic to Z2

They are also maximal elementary abelian subgroups of S.

S has no normal extra-special subgroups.

S has 511 involutions in 511 conjugacy classes.

All involutions are fused in G.

78 9 10 7.2 Realizable group of type L2(q), q ≡ 2 ± 1(mod 2 )

9 10 S is a Sylow-2 subgroup of simple groups L2(q), q ≡ 2 ± 1(mod 2 ). ∼ S = [512, 2042] = D512. ∼ 2 256 2 S = f1, f2 f1 = f2 = 1, [f1, f2] = f2

∼ Z(S) = Z2.

∼ ∼ 8 9 S/Z(S) = [256, 539] = D256 is a realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ). ∼ Note that [256,539] = D256 occurs among the maximal subgroups of S.

There are 3 maximal subgroups of S: [256, 539], [256, 539], [256, 1].

Two of them are realizable groups of lower order:

∼ 8 9 Both are [256, 539] = D256 of type L2(q), q ≡ 2 ± 1(mod 2 ). ∼ S/Φ(S) = Z2 × Z2. ∼ Maximal quotient group of S is isomorphic to [256, 539] = D256.

There is 1 maximal abelian subgroup of S.

Abelian Group A Z256 Number of groups in each conjugacy class 1 ∼ If G = L2(511),L2(513) then

NG(A)/CG(A) is isomorphic to Z2

There are 128 elementary abelian subgroups of S in 2 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 Number of groups in each conjugacy class 64+64 ∼ If G = L2(511),L2(513) then

NG(A)/CG(A) is isomorphic to: S3

79 There is no extra fusion that occurs in G for the maximal elementary abelian subgroups.

S has no normal extra-special subgroups.

S has 257 involutions in 3 conjugacy classes.

S S 128 S Conjugacy class (f1) (f1f2) (f2 ) Number of involutions 128 128 1

All involutions are fused in G.

7 8 7.3 Realizable group of type L3(q), q ≡ 2 −1(mod 2 ),U3(q), q ≡ 27 + 1(mod 28)

7 8 S is a Sylow-2 subgroup of simple groups L3(q), q ≡ 2 −1(mod 2 ) and U3(q), q ≡

27 + 1(mod 28). ∼ S = [512, 2043] = SD512.

∼ 2 256 27−2 S = f1, f2 f1 = f2 = 1, [f2, f1] = f2

∼ Z(S) = Z2.

∼ ∼ 8 9 S/Z(S) = [256, 539] = D256 is a realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ). ∼ Note that [256,539] = D256 occurs among the maximal subgroups of S.

There are 3 maximal subgroups of S: [256, 539], [256, 541] and [256, 1].

One of them is realizable group of lower order:

∼ 8 9 [256, 539] = D256 of type L2(q), q ≡ 2 ± 1(mod 2 ).

80 ∼ S/Φ(S) = Z2 × Z2. ∼ Maximal quotient group of S is isomorphic to [256, 539] = D539.

There is 1 maximal abelian subgroup of S.

Abelian Group A Z256 Number of groups in each conjugacy class 1 ∼ If G = L3(127),U3(129) then

NG(A)/CG(A) is isomorphic to Z2

There are 64 elementary abelian subgroups of S in 1 conjugacy class.

Elementary Abelian Group A Z2 × Z2 Number of groups in each conjugacy class 64 ∼ If G = L3(127),U3(129) then

NG(A)/CG(A) is isomorphic to S3

There is no extra fusion that occurs in G for the maximal elementary abelian subgroups.

S has no normal extra-special subgroups.

S has 129 involutions in 2 conjugacy classes.

S 128 S Conjugacy class (f1) (f2 ) Number of involutions 128 1

4 5 7.4 Realizable group of type L3(q), q ≡ 2 +1(mod 2 ),U3(q), q ≡ 24 − 1(mod 25)

4 5 S is a Sylow-2 subgroup of simple groups L3(q), q ≡ 2 + 1(mod 2 ),U3(q), q ≡

24 − 1(mod 25).

81 ∼ S = [512, 947] = Z2 × Z2 × Z2 × Z2 o Z2.

2 2 2 2 2 2 2 f1, f2, f3 f1 = f2, f2 = f3, f3 = f4, f4 = f5 = f6 = f9 = 1, ∼ 2 2 S = f4, f5, f6, f7 = f8, f8 = f9, [f5, f2] = [f6, f2] = [f6, f5] = f7f8f9,

f7, f8, f9 [f5, f4] = [f6, f4] = [f8, f5] = [f8, f6] = f9,

[f5, f3] = [f6, f3] = [f7, f5] = [f7, f6] = f8f9, [f5, f1] = [f6, f1] = f5f6f7f8f9

∼ Z(S) = Z16.

∼ ∼ 5 6 S/Z(S) = [32, 18] = D32 is a realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ).

There are 3 maximal subgroups of S:[256, 447], [256, 5034] and [256, 39]. None of them are realizable groups. ∼ S/Φ(S) = Z2 × Z2.

The following is maximal quotient group of S: ∼ If z = f9 then S/ < z >= [256, 56].

There is 1 maximal abelian subgroup of S.

Abelian Group A Z16 × Z16 Number of groups in each conjugacy class 1 ∼ If G = L3(17),U3(15) then

NG(A)/CG(A) is isomorphic to S3

There are 9 elementary abelian subgroups of S in 2 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 Number of groups in each conjugacy class 1+8 ∼ If G = L3(17),U3(15) then

NG(A)/CG(A) is isomorphic to S3

82 ∼ If S = L3(17),U3(15) then there is no extra fusion that occurs in G for the maximal elementary abelian subgroups.

S has no normal extra-special subgroups.

S has 19 involutions in 3 conjugacy classes.

S S S Conjugacy class (f9) (f6) (f4) Number of involutions 1 16 2

All involutions are fused in G.

4 7.5 Realizable group of type L4(q), q ≡ 7(mod 2 ),U4(q), q ≡ 9(mod 24)

4 S is a Sylow-2 subgroup of simple group L4(q), q ≡ 7(mod 2 ),U4(q), q ≡ 9(mod

24). ∼ ∼ S = D16 o Z2 = [512, 60809].

2 2 2 2 2 2 2 f1, f2, f3 f1 = f4 = f5 = f6 = f8 = f9 = 1, f2 = f3f4f5f6f7f9, ∼ 2 2 S = f4, f5, f6, f3 = f5f6f7f8, f7 = f9, [f2, f1] = f3f5f7f9, [f4, f1] = f5f6f8f9,

f7, f8, f9 [f3, f1] = [f6, f1] = [f7, f1] = [f7, f2] = [f6, f5] = f8,

[f3, f2] = f4f6f7f9, [f4, f2] = f4f6f9, [f5, f2] = f4f5f7f8, [f6, f4] = f7f9 [f6, f2] = f4f5f9, [f8, f2] = [f7, f4] = [f7, f5] = [f7, f6] = f9, [f4, f3] = [f5, f3] = [f6, f3] = f5f6f7f8, [f5, f4] = f7f8f9

∼ Z(S) = Z2.

∼ 3 4 S/Z(S) = [256, 6661] is a realizable group of type S4(q), q ≡ 2 ± 1(mod 2 ).

Note that [256,6661] occurs among the maximal subgroups of S.

There are 7 maximal subgroups of S: [256, 6661], [256, 6661], [256, 5298], [256, 12955],

83 [256, 503], [256, 503] and [256, 517].

Two of them are realizable groups of lower order:

3 4 They are both isomorphic to [256, 6661] of type S4(q), q ≡ 2 ± 1(mod 2 ). ∼ S/Φ(S) = Z2 × Z2 × Z2.

Maximal quotient group of S is isomorphic to [256,6661].

There is 1 maximal abelian subgroup of S.

Abelian Group A Z8 × Z8 Number of groups in each conjugacy class 1 ∼ If G = L4(7),U4(9) then

NG(A)/CG(A) is isomorphic to D8

There are 16 elementary abelian subgroups of S in 3 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 4+4+8

If G L4(7),U4(9) then

NG(A)/CG(A) is isomorphic to S5+S5+(S3 × S3)

∼ If S = L4(7),U4(9) then there is no extra fusion that occurs in G for the maximal elementary abelian subgroups.

S has no normal extra-special subgroups.

S has 115 involutions in 10 conjugacy classes.

S S S −1 S S −1 S Conjugacy class (f9) (f6) (f1) (f1f7 f6) (f8) (f6f3 ) Number of involutions 1 32 8 8 2 16

84 −1 S S −2 S S Conjugacy class (f4f3 ) (f1f9) (f5f1f3 ) (f1f2f8f6) Number of involutions 16 8 8 16

There are 2 conjugacy classes of involutions in G.

The ﬁrst one consists of 4 conjugacy classes of involutions of S fused in G:

S S S −1 S Conjugacy class (f9) ∪ (f6) ∪ (f1) ∪ (f1f7 f6) Number of involutions 1+32+8+8

The second one consists of 6 conjugacy classes of involutions of S fused in G:

S −1 S −1 S Conjugacy class (f8) ∪ (f6f3 ) ∪ (f4f3 ) ∪ S −2 S S (f1f9) ∪ (f5f1f3 ) ∪ (f1f2f8f6) Number of involutions 2+16+16+8+8+16

3 7.6 Realizable group of type L5(q), q ≡ 3(mod 2 ),U5(q), q ≡ 5(mod 23)

3 S is a Sylow-2 subgroup of simple groups L5(q), q ≡ 3(mod 2 ),U5(q), q ≡ 5(mod

23). ∼ ∼ S = SD12 o Z2 = [512, 60833].

2 2 2 2 2 2 2 2 2 f1, f2, f3 f1 = f4 = f6 = f8 = f9 = 1, f2 = f5 = f8, f3 = f7 = f8f9, ∼ S = f4, f5, f6, [f2, f1] = f2f3f5f6f7f8, [f3, f1] = f2f3f5f6f7f8f9, [f4, f1] = f6,

f7, f8, f9 [f8, f1] = [f7, f3] = [f7, f5] = [f7, f6] = f9, [f3, f2] = f4f7f9,

[f5, f1] = f4f6f7, [f4, f2] = [f6, f2] = f4f5f6f9, [f5, f2] = [f5, f4] = [f7, f4] = [f6, f5] = f8, [f7, f2] = f4f5f6, [f5, f3] = f5f6f7f9, [f6, f3] = f5f6f7f8f9, [f7, f1] = f6f9

∼ Z(S) = Z2.

85 S/Z(S) ∼= [256, 6662] is not a realizable group.

Note that [256,6662] occurs among the maximal subgroups of S.

There are 7 maximal subgroups of S: [256, 6671], [256, 6662], [256, 5352],

[256, 12965], [256, 508], [256, 505], [256, 523].

None of them is a realizable group of lower order: ∼ S/Φ(S) = Z2 × Z2 × Z2.

Maximal quotient group of S is isomorphic to [256, 6662].

There is 1 maximal abelian subgroup of S.

Abelian Group A Z8 × Z8 Number of groups in each conjugacy class 1 ∼ If G = L5(3),U5(5) then

NG(A)/CG(A) is isomorphic to D8

There are 4 elementary abelian subgroups of S in 1 conjugacy class.

Elementary Abelian Group A Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 4 ∼ If G = L5(3),U5(5) then

NG(A)/CG(A) is isomorphic to S5

S has no normal extra-special subgroups.

S has 51 involutions in 6 conjugacy classes.

S −1 S S S −1 S S Conjugacy class (f1) (f6f5 f9) (f8) (f6) (f6f5 ) (f1) Number of involutions 1 8 2 16 8 16

86 There are 2 conjugacy classes of involutions in G.

The ﬁrst one consists of 2 conjugacy classes of involutions of S fused in G:

S −1 S Conjugacy class (f9) ∪ (f6f5 f9) Number of involutions 1+8

The second one consists of 4 conjugacy classes of involutions of S fused in G:

S S −1 S S Conjugacy class (f8) ∪ (f6) ∪ (f6f5 ) ∪ (f1) Number of involutions 2+16+8+16

2 3 7.7 Realizable group of type S6(q), q ≡ 2 ± 1(mod 2 )

∼ 2 S = [512, 7530110] is a Sylow-2 subgroup of simple groups S6(q), q ≡ 2 ± 1(mod

23).

2 2 2 2 2 2 2 f1, f2, f3 f1 = f2 = f3 = f9, f4 = f6 = f8f9, f5 = f7 = f8, ∼ 2 2 S = f4, f5, f6, f = f = 1, [f4, f1] = f4f5f8f9, [f5, f1] = f4f5f8, 8 9 f7, f8, f9 [f2, f1] = [f8, f1] = [f3, f2] = f9, [f6, f1] = f6f7f8f9,

[f6, f4] = f8f9, [f7, f5] = f8, [f7, f1] = f6f7f8

∼ Z(S) = Z2.

S/Z(S) ∼= [256, 53380] is not a realizable group.

Note that [256, 53380] does not occur among the maximal subgroups of S.

There are 31 maximal subgroups of S:

87 Maximal subgroup [256, 26558] [256, 26555] [256, 26361] [256, 53376] [256, 56028] Number of groups 18 6 3 3 1

None of them are realizable groups of lower order. ∼ S/Φ(S) = Z2 × Z2 × Z2 × Z2 × Z2.

Maximal quotient group of S is isomorphic to [256, 53380].

There are 27 maximal abelian subgroups of S in 18 conjugacy classes.

Abelian Group A Z2 × Z4 × Z4 Number of groups in each conjugacy class 1+1+1+1+1+1+1+1+1+ 2+2+2+2+2+2+2+2+2 ∼ If G = S6(3),S6(5) then

NG(A)/CG(A) is isomorphic to [48,48]

∼ For G = S6(3),S6(5) maximal abelian subgroups isomorphic to Z2 ×Z4 ×Z4 are fused

in G.

There are 60 elementary abelian subgroups of S in 27 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 × Z2 × Z2 Z2 × Z2 × Z2 × Z2 Number of groups in 1+1+1+2+2+2+2+2+2+ 1+1+1+2+2+ each conjugacy class 2+2+2+2+2+2+4+4+4 2+4+4+4 ∼ If G = S6(3),S6(5) then

NG(A)/CG(A) is isomorphic to [96, 70] [288, 1025]

∼ For G = S6(3),S6(5) maximal elementary abelian subgroups isomorphic to Z2 × Z2 ×

Z2 × Z2 with isomorphic NG(A)/CG(A) are fused in G. So there are 2 conjugacy

88 classes of maximal elementary abelian subgroups in G. The ﬁrst one consists of 18

conjugacy classes of maximal elementary abelian subgroups in S and has 39 groups in

it and the second one consists of 9 conjugacy classes of maximal elementary abelian

subgroups in S and has 21 groups in it.

There is 1 maximal normal extra-special subgroup of S.

Extra-special Group E [128, 2326] Number of groups in each conjugacy class 1 ∼ If G = S6(3),S6(5) then

NG(E)/CG(E) is isomorphic to A group of order 11520

S has 143 involutions in 24 conjugacy classes.

S There is 1 conjugacy class of central involutions: f9 .

S There is 1 conjugacy class of involutions with 2 involutions in the class: f8 .

There are 9 conjugacy classes of involutions with 4 involutions in each class:

−1 −1 −1 S −1 −1 −1 S −1 −1 −1 −1 S −1 −1 −1 S −1 −1 −1 S (f7 f3 f6 ) ,(f5 f3 f4 ) ,(f5f7 f3 f6 f4 ) ,(f7 f6 f2 ) ,(f5 f4 f2 )

−1 −1 −1 −1 S −1 −1 −1 −1 S −1 −1 −1 −1 S −1 −1 −1 −1 S (f5f7 f6 f2 f4 ) ,(f7 f3 f2 f6 ) ,(f5 f3 f2 f4 ) ,(f5 f7 f2f3f4 f6 ) . There are 13 conjugacy classes of involutions with 8 involutions in each class:

−1 −1 S −1 −1 −1 S −1 −1 −1 S −1 −1 −1 S −1 −1 −1 S (f8 f1 ) ,(f5 f3 f6 ) ,(f5f7 f3 f6 ) ,(f5f7 f3 f4 ) ,(f5 f6 f2 ) ,

−1 −1 −1 S −1 −1 −1 S −1 −1 −1 −1 S −1 −1 −1 −1 S (f5f7 f6 f2 ) ,(f5f7 f4 f2 ) ,(f5 f3 f2 f6 ) ,(f5f7 f3 f2 f6 ) ,

−1 −1 −1 −1 S −1 −1 S −1 −1 −1 S −1 −1 S (f5f7 f3 f2 f4 ) ,(f3 f1 ) ,(f8 f2 f1 ) ,(f8 f1f2f3 ) . There are 2 conjugacy classes of involutions in G.

The ﬁrst one consists of 3 conjugacy classes of involutions of S fused in G:

S S −1 −1 S Conjugacy class f9 ∪ (f2f1) ∪ (f8 f1 ) Number of involutions 1+2+8

89 The second one consists of 21 conjugacy classes of involutions of S fused in G:

−1 −1 −1 S −1 −1 −1 S −1 −1 −1 −1 S Conjugacy (f7 f3 f6 ) ∪ (f5 f3 f4 ) ∪ (f5f7 f3 f6 f4 ) ∪ −1 −1 −1 S −1 −1 −1 S −1 −1 −1 −1 S class (f7 f6 f2 ) ∪ (f5 f4 f2 ) ∪ (f5f7 f6 f2 f4 ) ∪ −1 −1 −1 −1 S −1 −1 −1 −1 S −1 −1 −1 −1 S (f7 f3 f2 f6 ) ∪ (f5 f3 f2 f4 ) ∪ (f5 f7 f2f3f4 f6 ) −1 −1 −1 S −1 −1 −1 S −1 −1 −1 S (f5 f3 f6 ) ∪ (f5f7 f3 f6 ) ∪ (f5f7 f3 f4 ) −1 −1 −1 S −1 −1 −1 S −1 −1 −1 S (f5 f6 f2 ) ∪ (f5f7 f6 f2 ) ∪ (f5f7 f4 f2 ) ∪ −1 −1 −1 −1 S −1 −1 −1 −1 S −1 −1 −1 −1 S (f5 f3 f2 f6 ) ∪ (f5f7 f3 f2 f6 ) ∪ (f5f7 f3 f2 f4 ) −1 −1 S −1 −1 −1 S −1 −1 S (f3 f1 ) ∪ (f8 f2 f1 ) ∪ (f8 f1f2f3 ) . Number of 4+4+4+4+4+4+4+4+4+8+8 involutions 8+8+8+8+8+8+8+8+8+8+8

2 7.8 Realizable group of type S6(2),A12,A13,O7(q), q ≡ 2 ±1(mod 23)

2 S is a Sylow-2 subgroup of simple groups S6(2),A12,A13,O7(q), q ≡ 2 ± 1(mod

23).

S ∼= [512, 406983].

2 2 2 2 2 2 2 2 2 * f1 = f2 = f3 = f4 = f5 = f6 = f7 = f8 = f9 = 1, + ∼ f1, f2, f3, f4, f5, S = [f4, f3] = f5, [f6, f3] = [f6, f5] = f8, [f6, f4] = [f7, f4] = f6f7, f6, f7, f8, f9 [f3, f1] = f2, [f8, f4] = [f9, f4] = f8f9, [f7, f5] = f9

∼ Z(S) = Z2 × Z2.

S/Z(S) ∼= [128, 1755] is not a realizable group.

There are 15 maximal subgroups of S:

Maximal subgroup [256, 6029] [256, 26531] [256, 6334] [256, 6331] [256, 53380] Number of groups 2 2 2 2 1

90 Maximal subgroup [256, 25886] [256, 25876] [256, 51978] [256, 5675] [256, 6074] Number of groups 1 1 1 1 1

Maximal subgroup [256, 6079] Number of groups 1

None of them are realizable groups. ∼ S/Φ(S) = Z2 × Z2 × Z2 × Z2.

The following are maximal quotient groups of S: ∼ If z = f8f9 then S/ < z >= [256, 16888]. ∼ If z = f2 then S/ < z >= [256, 26531]. ∼ If z = f8f2f9 then S/ < z >= [256, 26539].

There is 1 maximal abelian subgroup of S.

It is also maximal elementary abelian subgroup of S.

Abelian Group A Z2 × Z2 × Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 1+1 ∼ If G = S6(3))

then NG(A)/CG(A) is isomorphic to [168,42] ∼ If G = A(12),A(13))

then NG(A)/CG(A) is isomorphic to [648,704] ∼ If G = O7(3))

then NG(A)/CG(A) is isomorphic to A group of order 2520

There are 4 normal extra-special subgroups of S in 4 conjugacy classes in G if ∼ G = A12,A13,S6(2).

91 Extra-special Group E [32, 49] Number of groups in each conjugacy class 1+1+1+1 ∼ If G = S6(2) then

NG(E)/CG(E) is isomorphic to [576,8654]+[64,138]+[64,138]+[192,955] ∼ If G = A12,A13 then

NG(E)/CG(E) is isomorphic to [64,138]+[64,138]+[64,138]+[192,955] ∼ If G = O7(3) then

NG(E)/CG(E) is isomorphic to [64,138]+[192,955]+[192,955]+[576,8654]

S has 143 involutions in 28 conjugacy classes.

S S S There are 3 conjugacy classes of central involutions: (f2) ,(f8f9) ,(f8f2f9) .

There are 4 conjugacy classes of involutions with 2 involutions in each class:

S S S S (f9) ,(f1) ,(f2f9) ,(f8f1f9) .

There are 11 conjugacy classes of involutions with 4 involutions in each class:

S S S S S S S S S S (f7) ,(f5) ,(f8f7) ,(f6f7) ,(f2f7) ,(f5f2) ,(f1f9) ,(f8f2f7) ,(f6f2f7) ,(f6f1f7) ,

S (f8f5f6f1f7f5) .

There are 9 conjugacy classes of involutions with 8 involutions in each class:

S S S S S S S S (f8f2f3f8f2) ,(f4) ,(f8f2f3f8f2f9) ,(f1f7) ,(f5f1) ,(f2f4) ,(f8f1f7) ,(f1f4) ,

S (f5f1f4) .

S There is 1 conjugacy class of involutions with 16 involutions in the class: (f8f2f3f8f2f7) . ∼ If G = A13 then there are 3 conjugacy classes of involutions in G.

The ﬁrst one consists of 6 conjugacy classes of involutions of S fused in G:

S S S S S S Conjugacy class f9 ∪ (f7) ∪ (f5) ∪ (f8f2f3f8f2) ∪ (f2) ∪ (f1) Number of involutions 2+4+4+8+1+2

92 The second one consists of 12 conjugacy classes of involutions of S fused in G:

S S S S S Conjugacy class (f8f9) ∪ (f8f7) ∪ (f6f7) ∪ (f4) ∪ (f8f2f3f8f2f9) ∪ S S S S (f8f2f3f8f2f7) ∪ (f2f9) ∪ (f2f7) ∪ (f5f2) ∪ S S S (f1f9) ∪ (f1f7) ∪ (f5f1) Number of involutions 1+4+4+8+8+16+2+4+4+4+8+8

The third one consists of 10 conjugacy classes of involutions of S fused in G:

S S S S Conjugacy class (f8f2f9) ∪ (f8f2f7) ∪ (f6f2f7) ∪ (f2f4) ∪ S S S S (f8f1f9) ∪ (f8f1f7) ∪ (f6f1f7) ∪ (f8f5f6f1f7f5) ∪ S S (f1f4) ∪ (f5f1f4) Number of involutions 1+4+4+8+2+8+4+4+8+8

∼ If G = A12 then there are 3 conjugacy classes of involutions in G with the following number of involutions in each class: 2+4+4+8+1+2, 1+4+4+8+8+16+2+4+4+4+8+8,

1+4+4+8+2+8+4+4+8+8. ∼ If G = S6(2) then there are 4 conjugacy classes of involutions in G with the following number of involutions in each class: 8+4+2+1+4+8, 4+1+8+2+4+8+4+2+8,

1+4+2, 4+8+16+4+4+4+8+4+8+8.

7.9 Realizable group of type L3(8)

S is a Sylow-2 subgroup of simple group L3(8).

S = [512, n]. S has rank 6 and p-class 2, so 7532393 ≤ n ≤ 10481221.4

4The ID number n of S in Small Groups library in GAP is not known. The number of 2-groups of the same rank and p-class is too high to complete the comprehensive search for these groups.

93 2 2 2 2 2 2 2 2 2 * f1, f2, f3, f1 = f2 = f3 = f4 = f5 = f6 = f7 = f8 = f9 = 1, + ∼ S = f4, f5, f6, [f9, f2] = f8, [f5, f1] = [f4, f3] = f6f7, [f9, f1] = [f5, f2] = f6f8, .

f7, f8, f9 [f4, f1] = f6f7f8, [f4, f2] = [f5, f3] = f7, [f9, f3] = f7f8

∼ Z(S) = Z2 × Z2 × Z2. ∼ ∼ S/Z(S) = [64, 267] = Z2 ×Z2 ×Z2 ×Z2 ×Z2 ×Z2 is a realizable group of type L2(64).

There are 63 maximal subgroups of S: 14 are isomorphic to [256, 46157] and 49 are isomorphic to [256, 46361] . None of them are realizable groups. ∼ S/Φ(S) = Z2 × Z2 × Z2 × Z2 × Z2 × Z2.

The following are maximal quotient groups of S: ∼ If z = f6, f7, f8, f6f7, f7f8, f6f8, f6f7f8 then S/ < z >= [256, 55960].

There are 9 maximal abelian subgroup of S in 9 conjugacy classes.

Abelian Group A Z2 × Z2 × Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 1+1

NG(A)/CG(A) is isomorphic to A group of order 3528+ A group of order 3528

Abelian Group A Z4 × Z4 × Z4 Number of groups in each conjugacy class 1+1+1+1+1+1+1

NG(A)/CG(A) is isomorphic to [56,11]+[56,11]+[56,11]+[56,11]+ [56,11]+[56,11]+[56,11]

Two of them are maximal elementary abelian subgroups of S isomorphic to Z2 ×Z2 ×

Z2 × Z2 × Z2 × Z2 .

There are no normal extra-special subgroups of S.

S has 119 involutions in 21 conjugacy classes.

S S S S There are 7 conjugacy classes of central involutions: (f6) ,(f7) ,(f8) ,(f6f7) ,

94 S S S (f7f8) ,(f6f8) ,(f6f7f8) .

There are 14 conjugacy classes of involutions with 8 involutions in each class:

S S S S S S S S S S S (f1) ,(f2) ,(f3) ,(f4) ,(f5) ,(f9) ,(f1f2) ,(f2f3) ,(f1f3) ,(f1f2f3) ,(f4f5) ,

S S S (f5f9) ,(f4f9) ,(f4f5f9) .

All involutions are fused in G.

7.10 Realizable group of type U3(8)

S is a Sylow-2 subgroup of simple group U3(8).

S = [512, n]. S has rank 6 and p-class 2, so 7532393 ≤ n ≤ 10481221.5

2 2 2 2 2 2 2 2 2 * f1, f2, f3, f1 = f2 = f3 = f4 = f5 = f6 = f7 = f8 = f9 = 1, + ∼ S = f4, f5, f6, [f4, f1] = [f5, f3] = f7, [f5, f1] = f6f7, [f9, f1] = f7f8, .

f7, f8, f9 [f4, f2] = f6f7f8, [f9, f2] = [f4, f3] = f8, [f5, f2] = [f9, f3] = f6f8

∼ Z(S) = Z2 × Z2 × Z2. ∼ ∼ S/Z(S) = [64, 267] = Z2 ×Z2 ×Z2 ×Z2 ×Z2 ×Z2 is a realizable group of type L2(64).

There are 63 maximal subgroups of S isomorphic to [256, 46352]. None of them are realizable groups. ∼ S/Φ(S) = Z2 × Z2 × Z2 × Z2 × Z2 × Z2.

The following are maximal quotient groups of S: ∼ If z = f6, f7, f8, f6f7, f7f8, f6f8, f6f7f8 then S/ < z >= [256, 55968].

There are 9 maximal abelian subgroups of S in 9 conjugacy classes.

5The ID number n of S in Small Groups library in GAP is not known. The number of 2-groups of the same rank and p-class is too high to complete the comprehensive search for these groups.

95 Abelian Group A Z4 × Z4 × Z4 Number of groups in each conjugacy class 1+1+1+1+1+1+1+1+1

NG(A)/CG(A) is isomorphic to [56,11]

There is 1 maximal elementary abelian subgroup of S.

Elementary Abelian Group A Z2 × Z2 × Z2 Number of groups in each conjugacy class 1

NG(A)/CG(A) is isomorphic to Z7

There are no normal extra-special subgroups of S.

S has 119 involutions in 21 conjugacy classes.

S S S S There are 7 conjugacy classes of central involutions: (f6) ,(f7) ,(f8) ,(f6f7) ,

S S S (f7f8) ,(f6f8) ,(f6f7f8) .

There are 14 conjugacy classes of involutions with 8 involutions in each class:

S S S S S S S S S S S (f1) ,(f2) ,(f3) ,(f4) ,(f5) ,(f9) ,(f3f2) ,(f3f1) ,(f2f1) ,(f2f1f3) ,(f9f5) ,

S S S (f5f4) ,(f9f4) ,(f5f4f9) .

All involutions are fused in G.

7.11 Realizable group of type HS

S is a Sylow-2 subgroup of simple group HS.

S ∼= [512, 60329].

2 2 2 2 2 2 2 2 2 f1, f2, f3 f1 = f6, f2 = f5f6, f3 = f4 = f5 = f6 = f7 = f8 = f9 = 1, ∼ S = f4, f5, f6, [f2, f1] = [f3, f1] = f2f3f6, [f4, f1] = [f8, f1] = f4f8,

f7, f8, f9 [f7, f1] = f6f9, [f4, f2] = f4f6f7, [f5, f2] = [f6, f2] = f5f6,

96 [f7, f2] = f4f5f7f9, [f8, f2] = f4f6f7f9, [f4, f3] = f7f8f9, [f5, f3] = [f6, f3] = f5f6, [f7, f3] = [f8, f3] = f7f8, [f8, f4] = [f7, f5] = [f8, f6] = [f6, f4] = [f7, f4] = f9

∼ Z(S) = Z2.

S/Z(S) ∼= [256, 6331] is not a realizable group.

There are 7 maximal subgroups of S: [256, 26558], [256, 26064], [256, 6550], [256, 26547],

[256, 1518], [256, 6562], [256, 485] . None of them are realizable groups. ∼ S/Φ(S) = Z2 × Z2 × Z2.

The following is maximal quotient group of S: [256, 6331].

There is 1 maximal abelian subgroup of S.

Abelian Group A Z4 × Z4 × Z4 Number of groups in each conjugacy class 1

NG(A)/CG(A) is isomorphic to [168,42]

There are 8 elementary abelian subgroups of S in 3 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 2+4+2

NG(A)/CG(A) is isomorphic to S6+[48,48]+[48,48]

There is no extra fusion that occurs in G for maximal elementary abelian subgroups of S.

There are no normal extra-special subgroups of S.

S has 87 involutions in 9 conjugacy classes.

97 S S S S −2 S S Conjugacy class (f9) (f8) (f7f8) (f6) (f2 ) (f3) Number of involutions 1 16 8 4 2 16

S S S Conjugacy class (f3f4f8) (f2f3f7f8) (f1f5f7) Number of involutions 16 8 16

There are 2 conjugacy classes of involutions in G.

The ﬁrst one consists of 6 conjugacy classes of involutions of S fused in G:

S S S S −2 S S Conjugacy class (f9) ∪ (f8) ∪ (f7f8) ∪ (f6) ∪ (f2 ) ∪ (f3) Number of involutions 1+16+8+4+2+16

The second one consists of 6 conjugacy classes of involutions of S fused in G:

S S S Conjugacy class (f3f4f8) ∪ (f2f3f7f8) ∪ (f1f5f7) ∪ Number of involutions 16+8+16

7.12 Realizable group of type O0Nan

S is a Sylow-2 subgroup of simple group O0Nan.6

S = [512, 58362].

2 2 2 2 2 2 2 f1, f2, f3 f1 = f2 = f3 = f8 = f9 = 1, f4 = f6f9, f5 = f8, ∼ 2 2 S = f4, f5, f6, f6 = f7 = f9, [f5, f1] = [f7, f1] = [f5, f3] = [f5, f4] = f8,

f7, f8, f9 [f6, f1] = [f6, f2] = [f8, f2] = [f7, f3] = [f7, f4] = f9,

[f5, f2] = f7f8f9, [f4, f3] = f7, [f2, f1] = f4, [f3, f1] = f5, [f4, f1] = [f4, f2] = f6

6Although matrix and permutation representations of group G =∼ O0Nan are known, none of them can be used in GAP to compute in reasonable time the fusion of elements of S in G.

98 ∼ Z(S) = Z2.

S/Z(S) ∼= [256, 5084] is not a realizable group.

There are 7 maximal subgroups of S: [256, 5161], [256, 5161], [256, 402], [256, 402],

[256, 3288], [256, 24064] and [256, 345].

None of them are realizable groups. ∼ S/Φ(S) = Z2 × Z2 × Z2.

The following is maximal quotient group of S: [256, 5084].

There is 1 maximal abelian subgroup of S.

Abelian Group A Z4 × Z4 × Z4 Number of groups in each conjugacy class 1

There are 33 elementary abelian subgroups of S in 6 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 × Z2 Number of groups in each conjugacy class 1+4+4+8+8+8

There are no normal extra-special subgroups of S.

S has 87 involutions in 7 conjugacy classes.

S S S S S S S Conjugacy class (f9) (f8) (f6f7) (f3) (f2) (f2f3) (f1) Number of involutions 1 2 4 16 16 16 32

99 CHAPTER 8

REALIZABLE GROUPS OF ORDER 210

In this chapter we list the properties of Realizable Groups of order 210.

Notation 8.0.1 We use the following notation:

S = [2n, k] - k-th group of order 2n in the Small Group Library in GAP 4.

S is of type G if S ∈ Syl2(G).

In power-commutator presentation we use commutator formula [a, b] = a−1b−1ab for a, b ∈ G.

There are 49487365422 groups of order 210, 11 of them are realizable groups. Groups of order 210 are not enumerated in a library.

8.1 Realizable group of type L2(1024)

S is a Sylow-2 subgroup of simple groups L2(1024).

S is an elementary abelian group of order 1024. ∼ S = Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2.

100 2 2 2 2 2 * f1, f2, f3, f4, f1 = f2 = f3 = f4 = f5 = + ∼ 2 2 2 2 2 S = f5, f6, f7, f8, f6 = f7 = f8 = f9 = f10 = 1,

f9, f10 [fi, fj] = 1 ∀ i, j

∼ Z(S) = Z2.

S/Z(S) ∼= Id is not a realizable group.

There are 511 maximal subgroups of S isomorphic to [512, 10494213].

All of them are realizable groups of lower order isomorphic to:[512, 10494213] ∼=

Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2 × Z2 of type L2(512).

S/Φ(S) ∼= Id. ∼ Maximal quotient group of S is isomorphic to [512, 10494213] = Z2 × Z2 × Z2 × Z2 ×

Z2 × Z2 × Z2 × Z2 × Z2.

There are 511 maximal abelian subgroup of S.

Abelian Group A Z2 × Z2 × Z2 × Z2 × Z2×

A Z2 × Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 1

NG(A)/CG(A) is isomorphic to Z2

They are also maximal elementary abelian subgroups of S.

S has no normal extra-special subgroups.

S has 1023 involutions in 1023 conjugacy classes.

All involutions are fused in G.

101 10 11 8.2 Realizable group of type L2(q), q ≡ 2 ± 1(mod 2 )

10 11 S is a Sylow-2 subgroup of simple groups L2(q), q ≡ 2 ± 1(mod 2 ). ∼ S = D1024.

∼ 2 512 2 S = f1, f2 f1 = f2 = 1, [f1, f2] = f2

∼ Z(S) = Z2.

∼ ∼ 9 10 S/Z(S) = [512, 2042] = D512 is a realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ). ∼ Note that [512,2042] = D512 occurs among the maximal subgroups of S.

There are 3 maximal subgroups of S: [512, 2042], [512, 2042] and [512, 1].

Two of them are realizable groups of lower order:

∼ 9 10 Both are [512, 2042] = D512 of type L3(q), q ≡ 2 ± 1(mod 2 ). ∼ S/Φ(S) = Z2 × Z2. ∼ Maximal quotient group of S is isomorphic to [512, 2042] = D512.

There is 1 maximal abelian subgroup of S.

Abelian Group A Z512 Number of groups in each conjugacy class 1

NG(A)/CG(A) is isomorphic to Z2

102 There are 256 elementary abelian subgroups of S in 2 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 Number of groups in each conjugacy class 128+128 ∼ If G = L2(511),L2(513)

then NG(A)/CG(A) is isomorphic to: S3

S has no normal extra-special subgroups.

S has 513 involutions in 3 conjugacy classes.

S S 256 S Conjugacy class (f1) (f1f2) (f2 ) Number of involutions 256 256 1

All involutions are fused in G.

8 9 8.3 Realizable group of type L3(q), q ≡ 2 −1(mod 2 ),U3(q), q ≡ 28 + 1(mod 29)

8 9 S is a Sylow-2 subgroup of simple groups L3(q), q ≡ 2 −1(mod 2 ) and U3(q), q ≡

28 + 1(mod 29). ∼ S = SD1024.

∼ 2 512 28−2 S = f1, f2 f1 = f2 = 1, [f2, f1] = f2 .

∼ Z(S) = Z2.

∼ ∼ 9 10 S/Z(S) = [512, 2042] = D512 is a realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ). ∼ Note that [512, 2042] = D512 occurs among the maximal subgroups of S.

103 There are 3 maximal subgroups of S: [512, 2042], [512, 2044] and [512, 1]. ∼ One of them is realizable group of lower order: [512, 2042] = D512 of type L2(q), q ≡

29 ± 1(mod 210). ∼ S/Φ(S) = Z2 × Z2. ∼ Maximal quotient group of S is isomorphic to [512, 2042] = D512.

There is 1 maximal abelian subgroup of S.

Abelian Group A Z512 Number of groups in each conjugacy class 1 ∼ If G = L3(127),U3(129)

then NG(A)/CG(A) is isomorphic to Z2

There are 128 elementary abelian subgroups of S in 1 conjugacy class.

Elementary Abelian Group A Z2 × Z2 Number of groups in each conjugacy class 128 ∼ 6 6 If G = L3(q), q ≡ 2 − 1,U3(q), q ≡ 2 + 1

then NG(A)/CG(A) is isomorphic to S3

There is no extra fusion that occurs in G for the maximal elementary abelian subgroups.

S has no normal extra-special subgroups.

S has 257 involutions in 2 conjugacy classes.

S 256 S Conjugacy class (f1) (f2 ) Number of involutions 256 1

104 4 5 8.4 Realizable group of type D4(q), q ≡ 2 ±1(mod 2 ),G2(q), q ≡ 24 ± 1(mod 25)

4 5 S is a Sylow-2 subgroup of simple group D4(q), q ≡ 2 ± 1(mod 2 ),G2(q), q ≡

24 ± 1(mod 25).

2 2 2 2 2 2 2 2 f1, f2, f3, f4 f1 = f2 = f9 = f10 = 1, f3 = f5, f4 = f6, f5 = f7, f6 = f8, ∼ 2 2 S = f5, f6, f7, f8, f = f9, f = f10, [f3, f1] = f5f7f9, [f4, f1] = f6f8f10, 7 8 f9, f10 [f5, f1] = f7f9, [f6, f1] = f8f10, [f7, f1] = f9, [f8, f1] = f10,

[f3, f2] = f3f4f5f7f9, [f4, f2] = f3f4f6f8f10, [f5, f2] = f5f6f7f9, [f6, f2] = f5f6f8f10, [f7, f2] = f7f8f9, [f8, f2] = f7f8f10, [f9, f2] = [f10, f2] = f9f10

∼ Z(S) = Z2.

S/Z(S) ∼= [512, 30471] is not a realizable group.

Note that [512,30471] does not occur among the maximal subgroups of S.

There are 7 maximal subgroups of S:

Maximal subgroup [512, 399728] [512, 30599] [512, 58830] [512, 947] [512, 1062] Number of groups 1 1 2 2 1

Two of them are realizable groups of lower order:

4 5 4 5 Both are [512, 947] of type L3(q), q ≡ 2 + 1(mod 2 ),U3(q), q ≡ 2 − 1(mod 2 ). ∼ S/Φ(S) = Z2 × Z2 × Z2.

Maximal quotient group of S is isomorphic to [512, 30471].

There is 1 maximal abelian subgroup of S.

Abelian Group A Z16 × Z16 Number of groups in each conjugacy class 1

105 There are 128 elementary abelian subgroups of S in 5 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 × Z2 Number of groups in each conjugacy class 16+16+32+32+32

There are no normal extra-special subgroups of S.

S has 291 involutions in 7 conjugacy classes.

S S S S S S S Conjugacy class (f9f10) (f10) (f2) (f2f1) (f1) (f1f3f4) (f1f4) Number of involutions 1 2 16 16 64 64 128

All involutions are fused in G.

4 5 8.5 Realizable group of type S4(q), q ≡ 2 ± 1(mod 2 )

4 5 S is a Sylow-2 subgroup of simple groups S4(q), q ≡ 2 ± 1(mod 2 ).

S is of order 1024.

2 2 2 f1, f2, f3, f4 f1 = f2f3f4f6, f2 = f4f7f8f10, f3 = f5f6f9f10, ∼ 2 2 2 2 2 2 2 S = f5, f6, f7, f8, f4 = f5 = f6 = f7 = f = 1, f = f = f10, [f6, f1] = f4f6, 10 8 9 f9, f10 [f3, f1] = f2f3f5f6, [f4, f1] = f4f5f9, [f5, f1] = f5f7f8f10

[f2, f1] = f2f3f4f5f6f7, [f7, f1] = f6f7, [f8, f1] = f8f9, [f7, f4] = f8f10, [f5, f3] = [f6, f3] = f5f6f9f10, [f4, f2] = [f7, f2] = f4f7f8f10, [f6, f5] = f9f10, [f8, f4] = [f9, f5] = [f9, f6] = [f8, f7] = f10, [f9, f1] = f8f9f10

∼ Z(S) = Z2.

∼ 4 S/Z(S) = [512, 60809] is a realizable group of type L4(q), q ≡ 7(mod 2 ),U4(q), q ≡

9(mod 24).

Note that [512, 60809] occurs among the maximal subgroups of S.

106 There are 7 maximal subgroups of S.

Maximal subgroup [512, 2015] [512, 2019] [512, 60809] [512, 30471] [512, 399728] Number of groups 1 2 2 1 1

Two of them are realizable groups of lower order: [512, 60809] of type L4(q), q ≡

4 4 7(mod 2 ),U4(q), q ≡ 9(mod 2 ). ∼ S/Φ(S) = Z2 × Z2 × Z2.

Maximal quotient group of S is isomorphic to [512, 60809].

There is 1 maximal abelian subgroup of S.

Abelian Group A Z8 × Z16 Number of groups in each conjugacy class 1 ∼ If G = S4(17)

then NG(A)/CG(A) is isomorphic to D8

There are 16 elementary abelian subgroups of S in 2 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 8+8 ∼ If G = S4(17)

then NG(A)/CG(A) is isomorphic to S5

There are no normal extra-special subgroups of S.

S has 195 involutions in 9 conjugacy classes.

S −1 S −1 −1 S −1 S −1 −1 S Conjugacy class (f10) (f7f1f3 f10) (f8 f9 ) (f7 ) (f7 f6 ) Number of involutions 1 16 2 16 32

107 Conjugacy −1 −1 S −1 −1 −1 S −1 −1 −1 S −1 −1 S class (f6 f3 ) (f7 f6 f3 ) (f7 f2 f7 ) (f7 f1f3 ) Number of involutions 16 64 32 16

There are 2 conjugacy classes of involutions in G.

The ﬁrst one consists of 2 conjugacy classes of involutions of S fused in G:

S −1 S Conjugacy class (f10) ∪ (f7f1f3 f10) Number of involutions 1+16

The second one consists of 7 conjugacy classes of involutions of S fused in G:

−1 −1 S −1 S −1 −1 S −1 −1 S Conjugacy class (f8 f9 ) ∪ (f7 ) ∪ (f7 f6 ) ∪ (f6 f3 ) ∪ −1 −1 −1 S −1 −1 −1 S −1 −1 S (f7 f6 f3 ) ∪ (f7 f2 f7 ) ∪ (f7 f1f3 ) ∪ Number of involutions 2+16+32+16+64+32+16

8.6 Realizable group of type He, M24,L5(2)

S is a Sylow-2 subgroup of simple groups He, M24,L5(2).

S is of order 1024.

2 2 2 2 2 2 2 2 2 2 f1, f2, f3, f4 f1 = f2 = f3 = f4 = f5 = f6 = f7 = f8 = f9 = f10 = 1, ∼ S = f5, f6, f7, f8, [f2, f1] = f2f3f4f6f10, [f3, f1] = f2f3f5f6f8f9, [f4, f1] = f4f5f9f10,

f9, f10 [f6, f1] = f8f10, [f7, f1] = [f9, f1] = [f3, f2] = f10, [f4, f2] = f5f6,

[f5, f2] = [f6, f2] = f5f6f10, [f7, f2] = [f8, f2] = [f9, f2] = f8f9f10, [f5, f1] = f4f5f9, [f4, f3] = [f6, f3] = f4f6, [f5, f3] = f4f6f10, [f8, f3] = [f9, f3] = f8f9, [f7, f4] = f9, [f7, f5] = f9f10, [f7, f6] = f8

∼ Z(S) = Z2.

108 S/Z(S) ∼= [512, 152421] is not a realizable group.

Note that [512,152421] does not occur among the maximal subgroups of S.

There are 15 maximal subgroups of S.

Maximal subgroup [512, 254238] [512, 418986] [512, 46939] [512, 46946] [512, 51574] Number of groups 2 2 2 2 2

Maximal subgroup [512, 6407070] [512, 41035] [512, 51560] [512, 51562] Number of groups 2 1 1 1

None of them is a realizable group of lower order. ∼ S/Φ(S) = Z2 × Z2 × Z2 × Z2.

Maximal quotient group of S is isomorphic to [512, 152421].

There are 2 maximal abelian subgroups of S in 2 conjugacy classes.

Abelian Group A Z2 × Z2 × Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 1+1 ∼ If G = M24

then NG(A)/CG(A) is isomorphic to [1008,883]

These 2 maximal abelian subgroups are also elementary abelian subgroups of S.

There is 1 maximal normal extra-special subgroup E of S isomorphic to [128, 2326]

NG(E)/CG(E) is isomorphic to a group of order 10752.

S has 191 involutions in 25 conjugacy classes.

S There is 1 conjugacy class with 1 central involution: (f10) .

S There are 3 conjugacy classes of involutions with 2 involutions in each class: (f9f8) ,

109 S S (f4f5) ,(f9f8f4f5) .

S There are 6 conjugacy classes of involutions with 4 involutions in each class: (f3f8f3) ,

S S (f3f8f6f8f4f8f3f4f5) ,(f3f8f6f8f4f9f8f4f2f6) ,

S S S (f3f8f6f5f3) ,(f3f8f6f8f2f6) ,(f3f8f6f8f4f8f3f6f5) .

S There are 10 conjugacy classes of involutions with 8 involutions in each class: (f7) ,

S S S S S (f3f8f6f8f4f8f3f4) ,(f6f5) ,(f7f3) ,(f8f7f3f8f3f1) ,(f3f8f6f8f4f8f3f4f10) ,

S S S S (f7f3f10) ,(f3f8f2) ,(f7f3f6f2f6) ,(f9f7f1) .

S There are 5 conjugacy classes of involutions with 16 involutions in each class: (f6) ,

S S S S (f3) ,(f9f7f6f5f3) ,(f1) ,(f3f8f6f8f4f8f7f3f8f4f1) . ∼ If G = M24 then there are 2 conjugacy classes of involutions in G.

The ﬁrst one consists of 13 conjugacy classes of involutions in S that are fused in G:

S S S S Conjugacy class (f10) ∪ (f3f8f3) ∪ (f9f8) ∪(f7) ∪ S S S (f6) ∪ (f3f8f6f8f4f8f3f4) ∪(f6f5) ∪ S S S S (f3f8f6f8f4f8f3f4f5) ∪ (f4f5) ∪(f3) ∪ (f7f3) ∪ S S (f3f8f6f8f4f9f8f4f2f6) ∪ (f8f7f3f8f3f1) Number of involutions 1+4+2+8+16+8+8+4+2+16+8+4+8

The second one consists of 12 conjugacy classes of involutions of S fused in G:

S S Conjugacy class (f3f8f6f8f4f8f3f4f10) ∪ (f3f8f6f5f3) ∪ S S (f3f8f6f8f4f8f3f4f5) ∪ (f9f8f4f5) ∪ S S S (f7f3f10) ∪ (f9f7f6f5f3) ∪ (f3f8f2) ∪ S S S (f3f8f6f8f2f6) ∪ (f7f3f6f2f6) ∪ (f1) ∪ S S (f9f7f1) ∪ (f3f8f6f8f4f8f7f3f8f4f1) Number of involutions 8+4+4+2+8+16+8+4+8+16+8+16

110 ∼ If G = L5(2) then there are 2 conjugacy classes of involutions in G.

The ﬁrst one consists of 10 conjugacy classes of involutions in S with the following number of involutions: 1+2+2+4+4+4+8+8+8+8 that are fused in G.

The second one consists of 15 conjugacy classes of involutions in S with the following number of involutions: 2+4+4+4+8+8+8+8+8+8+16+16+16+16+16 that are fused in G.

8.7 Realizable group of type Co3

S is a Sylow-2 subgroup of simple groups Co3.

2 2 2 2 2 2 2 2 2 f1, f2, f3, f4 f1 = f3, f2 = f3 = f4 = f6 = f7 = f8 = f9 = f10 = 1, ∼ S = f5, f6, f7, f8, [f2, f1] = f3f6f8f9f10, [f4, f1] = f3f5f7f8, [f5, f1] = f5f7f8f9,

f9, f10 [f6, f1] = [f7, f1] = f6f7f10, [f8, f1] = f5f6f8f10, [f5, f2] = f9,

[f4, f2] = f3f5f6f7f9f10, [f3, f2] = f5f7f8f9f10, [f6, f2] = f5f9f10, [f7, f2] = f7f8f10, 2 f5 = f9, [f8, f2] = f7f8f9, [f4, f3] = [f5, f3] = [f8, f3] = [f5, f4] = [f6, f5] = f9, [f9, f1] = [f6, f3] = [f7, f3] = [f7, f6] = [f8, f6] = f10, [f6, f4] = [f8, f7] = f9f10

∼ Z(S) = Z2. ∼ S/Z(S) = [512, 406983] is a realizable group of type S6(2), A12, A13, O7(q), q ≡

22 ± 1(mod 23).

Note that [512, 406983] does not occur among the maximal subgroups of S.

There are 15 maximal subgroups of S: [512, 254142], [512, 59941], [512, 420081],

[512, 60375], [512, 420096], [512, 60345], [512, 419034], [512, 7530110], [512, 59903],

[512, 59223], [512, 60329], [512, 59766], [512, 60321], [512, 59753] and [512, n], where

[512, n] has rank 5 and p-class 3, so 6269624 ≤ n ≤ 7529606.7

7The ID number n in Small Groups library in GAP is not known. The number of 2-groups of the same rank and p-class is too high to complete the comprehensive search for these groups.

111 None of them are realizable groups of lower order. ∼ S/Φ(S) = Z2 × Z2 × Z2 × Z2 × Z2.

Maximal quotient group of S is isomorphic to [512, 406983].

There is 1 maximal abelian subgroup of S.

Abelian Group A Z4 × Z4 × Z4 Number of groups in each conjugacy class 1

NG(A)/CG(A) is isomorphic to [336, 209]

There are 2 elementary abelian subgroups of S in 1 conjugacy class.

Elementary Abelian Group A Z2 × Z2 × Z2 × Z2 Z2 × Z2 × Z2 × Z2 Number of groups in 1+2+2+2+4+4+ 1+2 each conjugacy class 4+4+4+4+4+8

Elementary Abelian Group A Z2 × Z2 × Z2 × Z2 Z2 × Z2 × Z2 × Z2 Number of groups in 2+4+4+8 8 each conjugacy class

There is 1 maximal normal extra-special subgroup E of S isomorphic to [128, 2326].

NG(E)/CG(E) is isomorphic to a group of order 10752.

S has 191 involutions in 17 in conjugacy classes.

S S S −1 S S Conjugacy class (f10) (f9) (f4) (f5 f4f8) (f3) Number of involutions 1 2 16 8 8

S S S S S Conjugacy class (f5f8f6f3) (f3f4f1f5) (f8) (f8f4) (f4f7) Number of involutions 4 16 16 16 8

112 S S S S Conjugacy class (f9f4f8f6f7) (f5f4) (f8f7f3) (f2) Number of involutions 8 16 8 32

−1 S S −1 S Conjugacy class (f9f8f2f1 ) (f4f3f5f2f1) (f4f3f2f1 ) Number of involutions 16 8 8

There are 2 conjugacy classes of involutions in G.

The ﬁrst one consists of 7 conjugacy classes of involutions of S fused in G:

S S S −1 S S Conjugacy class (f10) ∪ (f9) ∪ (f4) ∪ (f5 f4f8) ∪ (f3) ∪ S S (f5f8f6f3) ∪ (f3f4f1f5) Number of involutions 1+2+16+8+8+4+16

The second one consists of 10 conjugacy classes of involutions of S fused in G:

S S S S Conjugacy class (f8) ∪ (f8f4) ∪ (f4f7) ∪ (f9f4f8f6f7) ∪ S S S −1 S (f5f4) ∪ (f8f7f3) ∪ (f2) ∪ (f9f8f2f1 ) ∪ S −1 S (f4f3f5f2f1) ∪ (f4f3f2f1 ) Number of involutions 16+16+8+8+16+8+32+16+8+8

4 8.8 Realizable group of type L4(q), q ≡ 9(mod 2 ),U4(q), q ≡ 7(mod 24)

4 S is a Sylow-2 subgroup of simple groups L4(q), q ≡ 9(mod 2 ),U4(q), q ≡ 7(mod

24).

113 2 2 2 2 2 2 2 2 f1, f2, f3, f4 f1 = f5f7f8f9f10, f2 = f3 = f4 = f5 = f6 = f7 = f10 = 1, ∼ 2 2 S = f5, f6, f7, f8, f8 = f9 = f10, [f2, f1] = f2f3f4f5f6f7f9, [f3, f1] = f2f3f10,

f9, f10 [f4, f1] = f4f6f9, [f5, f1] = f5f7f10, [f6, f1] = f4f6, [f7, f1] = f5f7,

[f8, f1] = f8f9, [f3, f2] = [f9, f5] = [f9, f6] = [f8, f7] = [f8, f4] = f10, [f4, f2] = f4f5, [f5, f2] = f4f5, [f6, f2] = f6f7f8f10, [f7, f2] = f6f7f9, [f9, f1] = f8f9f10, [f8, f2] = [f9, f2] = f8f9f10, [f4, f3] = [f5, f3] = f4f5f10, [f6, f3] = [f7, f3] = f6f7,

[f8, f3] = [f9, f3] = f8f9, [f7, f4] = f8, [f6, f5] = f9f10

∼ Z(S) = Z2.

S/Z(S) ∼= [512, 60313] is not a realizable group.

Note that [512, 60313] does not occur among the maximal subgroups of S.

There are 7 maximal subgroups of S.

Maximal subgroup [512, 39030] [512, 250295] [512, 60645] [512, 1995] [512, 420056] Number of groups 1 1 2 1 2

None of them are realizable groups. ∼ S/Φ(S) = Z2 × Z2 × Z2.

Maximal quotient group of S is isomorphic to [512, 60313].

There is 1 maximal abelian subgroup of S.

Abelian Group A Z2 × Z8 × Z8 Number of groups in each conjugacy class 1

NG(A)/CG(A) is isomorphic to S4

114 There are 16 elementary abelian subgroups of S in 4 conjugacy classes.

Elementary Abelian Group A Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 4+4+4+4

There is 1 normal extra-special subgroup of S isomorphic to [128, 2326].

NG(E)/CG(E) is isomorphic to: A group of order 10752.

S has 135 involutions in 9 in conjugacy classes.

S −1 −1 S S S S Conjugacy class (f10) (f9 f8 ) (f7f6) (f7f5) (f3) Number of involutions 1 2 16 16 32

S −1 −1 S S −1 S Conjugacy class (f2f3f5) (f7f9 f1 ) (f7) (f3f2f4f5f9 ) Number of involutions 16 32 16 4

There are 2 conjugacy classes of involutions in G.

The ﬁrst one consists of 7 conjugacy classes of involutions of S fused in G:

S −1 −1 S S −1 S Conjugacy class (f10) ∪ (f9 f8 ) ∪ (f7f6) ∪ (f7f5 f4f8) ∪ S S −1 S (f3) ∪ (f2f3f5) ∪ (f3f2f4f5f9 ) Number of involutions 1+2+16+16+32+16+32

The second one consists of 2 conjugacy classes of involutions of S fused in G:

S −1 S Conjugacy class (f7) ∪ (f3f2f4f5f9 ) Number of involutions 16+4

115 8.9 Realizable group of type U5(2)

S is a Sylow-2 subgroup of simple group U5(2).

2 2 2 2 2 2 2 f1, f2, f3, f4 f5 = f6 = f7 = f8 = f9 = f10, f1 = f2 = f4, ∼ 2 2 2 S = f5, f6, f7, f8, f3 = f4f5f6f10, f4 = f10 = 1, [f2, f1] = f4, [f3, f1] = f8f9f10,

f9, f10 [f5, f1] = f5f9, [f6, f1] = f6f8, [f7, f1] = [f8, f1] = f6f7f10,

[f9, f1] = f5f7f8f9, [f3, f2] = f4f6f7f8f9f10, [f5, f2] = f5f6f7f8, [f6, f2] = f5f6, [f7, f2] = [f8, f2] = f7f9f10, [f9, f2] = f8f9, [f4, f3] = f5f6f8f9f10, [f5, f4] = [f6, f4] = [f7, f4] = [f8, f4] = [f9, f4] = f7f8f10, [f6, f5] = [f7, f5] = [f8, f5] =

[f9, f5] = [f7, f6] = [f8, f6] = [f9, f6] = [f9, f7] = [f9, f8] = f10, [f7, f3] = f6f8f10, [f8, f3] = f6f8, [f9, f3] = f5f9f10, [f6, f3] = f6f7f10, [f5, f3] = f5f7f8f9f10

∼ Z(S) = Z2.

S/Z(S) ∼= [512, 253472] is not a realizable group.

Note that [512, 253472] does not occur among the maximal subgroups of S.

There are 15 maximal subgroups of S.

Maximal subgroup [512, 41038] [512, 51849] [512, 51825] Number of groups 3 9 3

None of them are realizable groups. ∼ S/Φ(S) = Z2 × Z2 × Z2 × Z2 × Z2.

Maximal quotient group of S is isomorphic to [512, 253472].

There are 15 maximal abelian subgroups of S of order 32 in 5 conjugacy classes.

116 Abelian Group A Z2 × Z2 × Z2 × Z4 Number of groups in each conjugacy class 1+2+4+4+4

NG(A)/CG(A) is isomorphic to [96, 204]

There is 1 maximal elementary abelian subgroup of S in 1 conjugacy class.

Elementary Abelian Group A Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 1

There is 1 normal extra-special subgroup E of S isomorphic to [128, 2327]

NG(E)/CG(E) is isomorphic to a group of order 41472.

S has 63 involutions in 9 in conjugacy classes.

S S −1 −1 −1 −1 S Conjugacy class (f10) (f4) (f5 f7 f5 f8 ) Number of involutions 1 4 2

−1 −1 −1 −1 −2 −1 −1 −1 −1 −1 S Conjugacy class (f4f5 f7 f5 f8 f3 f5 f7 f5 f9 f5 ) Number of involutions 16

−1 −1 −1 −1 −2 −1 −1 −1 −1 S Conjugacy class (f10f4f5 f7 f5 f8 f3 f5 f7 f5 f9 ) Number of involutions 2

−2 −1 −1 −1 −1 S S −1 −1 −1 S Conjugacy class (f4f3 f5 f7 f5 f9 ) (f10f4) (f10f1 f5 f3 ) Number of involutions 2 4 16

−1 −1 −1 −1 −1 −1 S Conjugacy class (f5 f7 f5 f8 f1 f3 ) Number of involutions 16

117 There are 2 conjugacy classes of involutions in G.

The ﬁrst one consists of 2 conjugacy classes of involutions of S fused in G:

S S Conjugacy class (f10) ∪ (f4) Number of involutions 1+4

The second one consists of 7 conjugacy classes of involutions of S fused in G:

−1 −1 −1 −1 S Conjugacy (f5 f7 f5 f8 ) ∪ −1 −1 −1 −1 −2 −1 −1 −1 −1 −1 S class (f4f5 f7 f5 f8 f3 f5 f7 f5 f9 f5 ) ∪ −1 −1 −1 −1 −2 −1 −1 −1 −1 S (f10f4f5 f7 f5 f8 f3 f5 f7 f5 f9 ) ∪ S −1 −1 −1 S −1 −1 −1 −1 −1 −1 S (f10f4) ∪ (f10f1 f5 f3 ) ∪ (f5 f7 f5 f8 f1 f3 ) Number of involutions 2+16+2+2+4+16+16

8.10 Realizable group of type Sz(32)

S is a Sylow-2 subgroup of simple group Sz(32).

2 2 2 2 2 2 2 f1, f2, f3, f4 f1 = f2 = f3 = f4 = f5 = 1, f6 = f1, f7 = f1f2f5, ∼ 2 2 = S = f5, f6, f7, f8, f8 = f4, f9 = f3f4, f10 f2f4, [f7, f6] = f1f4f5, −1 −1 f9, f10 [f8, f6] = f2f3, [f9, f6] = f7 f4f7 , [f10, f6] = f3f5,

[f8, f7] = f1f2f3, [f9, f7] = f1f3, [f10, f7] = f3, [f9, f8] = f2f3f5, [f10, f8] = f1f3f5, [f10, f9] = f1f3f4f5

∼ Z(S) = Z2 × Z2 × Z2 × Z2 × Z2. ∼ ∼ S/Z(S) = [32, 51] = Z2 × Z2 × Z2 × Z2 × Z2 is a realizable group of type L2(32).

118 There are 31 maximal subgroups of S isomorphic to [512, 67607]. None of them are realizable groups. ∼ S/Φ(S) = Z2 × Z2 × Z2 × Z2 × Z2 × Z2.

There are 31 maximal quotient groups of S isomorphic to [512, ni] which has rank 5

8 and p-class 2, so 420515 ≤ ni ≤ 6249623 for 1 ≤ i ≤ 31.

There are 31 maximal abelian subgroups of S in 31 conjugacy classes.

Abelian Group A Z2 × Z2 × Z2 × Z2 × Z4 Number of groups in 1+1+1+1+1+1+1+1+1+1 each conjugacy class 1+1+1+1+1+1+1+1+1+1 1+1+1+1+1+1+1+1+1+1+1

NG(A)/CG(A) is isomorphic to E16

There is 1 maximal elementary abelian subgroup of S in 1 conjugacy class.

Elementary Abelian Group A Z2 × Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 1

NG(A)/CG(A) is isomorphic to Z31

There are no normal extra-special subgroups of S.

S has 31 involution in 31 conjugacy classes. All involutions are central involutions.

All involutions are fused in G.

8 The ID numbers ni in Small Groups library in GAP are not known. The number of 2-groups of the same rank and p-class is too high to complete the comprehensive search for these groups.

119 − 2 8.11 Realizable group of type A14,A15,O8 (q), q ≡ 2 ± 3(mod 23)

− 2 3 S is a Sylow-2 subgroup of simple group A14,A15,O8 (q), q ≡ 2 ± 3(mod 2 ).

2 2 2 2 2 2 2 2 2 2 * f1, f2, f3, f1 = f2 = f3 = f4 = f5 = f6 = f7 = f8 = f9 = f10 = + ∼ S = f4, f5, f6, 1, [f2, f1] = [f4, f2] = f3, [f7, f4] = f9, [f7, f5] = [f8, f5] = f7f8,

f7, f8, f9, f10 [f5, f4] = f6, [f9, f5] = [f10, f5] = f9f10, [f7, f6] = f9, [f8, f6] = f10

∼ Z(S) = Z2 × Z2.

S/Z(S) ∼= [256, 53380] is not a realizable group.

There are 31 maximal subgroups of S.

Maximal subgroup [512, 404483] [512, 406983] [512, 418967] [512, 407039] Number of groups 2 4 2 2

Maximal subgroup [512, 408352] [512, 410378] [512, 407953] [512, 404495] Number of groups 6 2 2 1

Maximal subgroup [512, 407965] [512, 410390] [512, 402937] [512, 419104] Number of groups 1 1 1 1

Maximal subgroup [512, 7530050] [512, n1] [512, n2] [512, n3] [512, n4] Number of groups 2 1 1 1 1

Where [512, n1], [512, n2], [512, n3] have rank 5 and p-class 3, so 6269624 ≤ n1, n2, n3 ≤

9 7529606 and [512, n4] has rank 6 and p-class 2, so 7532393 ≤ n4 ≤ 10481221.

Four of them are realizable groups isomorphic to [512, 406983] of type S6(2), A12, A13,

9 The ID numbers n1, n2, n3, n4 in Small Groups library in GAP are not known. The number of 2-groups of the same rank and p-class is too high to complete the comprehensive search for these groups.

120 2 3 O7(q), q ≡ 2 ± 1(mod 2 ). ∼ S/Φ(S) = Z2 × Z2 × Z2 × Z2 × Z2.

Maximal quotient groups of S are: [512, 7530050],[512, 7530076] and [512, n], where

[512, n] has rank 5 and p-class 3, so 6269624 ≤ n ≤ 7529606.

There are 27 maximal abelian subgroups of S in 18 conjugacy classes.

Abelian Group A Z4 × Z4 Z2 × Z2 × Z4 × Z4 Number of groups in each conjugacy class 1 1+1+2+2 ∼ If G = A14 [48, 38]+[48, 38]+

then NG(A)/CG(A) is isomorphic to S4 × Z2 [48, 38]+[48, 38]

Abelian Group A Z2 × Z2 × Z2 × Z2 × Z4 Number of groups in each conjugacy class 1+2+1+2+2+2+2 ∼ If G = A14 S5 × Z2 + S5 × Z2+

then NG(A)/CG(A) is isomorphic to [144, 186] + [144, 186]+

S3 × S3 × Z2+S3 × S3 × Z2

S3 × S3 × Z2

Abelian Group A Z2 × Z2 × Z2 × Z2 × Z2 × Z2 Number of groups in each conjugacy class 1+2+1+1+2+1 ∼ If G = A14 A group of order 5040+

then NG(A)/CG(A) is isomorphic to [432, 741] + [432, 741] + [1296, 3490]+

S5 × S3 + S5 × S3

There are 8 elementary abelian subgroups of S in 6 conjugacy classes. They are listed among maximal abelian subgroups.

There is no extra fusion that occurs in G for maximal elementary abelian subgroups of S.

There are no normal extra-special subgroups of S.

121 S has 263 involutions in 43 conjugacy classes.

S There are 3 conjugacy classes with 1 central involution in each class: (f4f3f1f4f1) ,

S S (f9f10) ,(f4f3f1f9f4f1f10) .

There are 6 conjugacy classes of involutions with 2 involutions in each class:

S S S S S S (f10) ,(f2) ,(f1) ,(f4f3f1f4f1f10) ,(f9f2f10) ,(f9f1f10) .

S There are 14 conjugacy classes of involutions with 4 involutions in each class: (f8) ,

S S S S S S S S (f6) ,(f4) ,(f4f3f1) ,(f9f8) ,(f7f8) ,(f4f3f1f4f1f8) ,(f4f3f1f6f4f1) ,(f2f10) ,

S S S S S S (f1f10) ,(f4f1) ,(f4f3f1f10) ,(f4f3f1f9f4f1f8) ,(f4f3f1f9f7f4f1f8) ,(f4f1f10) .

There are 16 conjugacy classes of involutions with 8 involutions in each class:

S S S S S S S S S (f8) ,(f5) ,(f4f10) ,(f2f8) ,(f6f2) ,(f1f8) ,(f6f1) ,(f4f3f1f8) ,(f4f3f1f2) ,

S S S S S S S (f4f3f1f4f1f5) ,(f9f2f8) ,(f7f2f8) ,(f9f1f8) ,(f7f1f8) ,(f4f1f8) ,(f4f3f1f2f10) .

There are 4 conjugacy classes of involutions with 16 involutions in each class:

S S S S (f4f8) ,(f2f5) ,(f1f5) ,(f4f3f1f2f8) . ∼ If G = A14 then there are 3 conjugacy classes of involutions in G.

The ﬁrst one consists of 8 conjugacy classes of involutions in S that are fused in G:

S S S S S Conjugacy class (f10) ∪ (f8) ∪ (f6) ∪ (f4) ∪(f7) ∪ S S S S (f4f3f1f4f1) ∪ (f2) ∪(f1) ∪ (f4f3f1) Number of involutions 2+4+4+8+1+2+2+4

122 The second one consists of 19 conjugacy classes of involutions of S fused in G:

S S S S Conjugacy class (f9f10) ∪ (f9f8) ∪ (f7f8) ∪ (f5) ∪ S S S (f4f10) ∪ (f4f8) ∪ (f4f3f1f4f1f10) ∪ S S S (f4f3f1f4f1f8) ∪ (f4f3f1f6f4f1) ∪ (f2f10) ∪ S S S S (f2f8) ∪ (f6f2) ∪ (f1f10) ∪ (f1f8) ∪ S S S (f6f1) ∪ (f4f1) ∪ (f4f3f1f10) ∪ S S (f4f3f1f8) ∪ (f4f3f1f2) Number of involutions 1+4+4+8+8+16+2+4+4+4+8+8+4+8+8+4+4+8+8

The third one consists of 16 conjugacy classes of involutions of S fused in G:

S S Conjugacy class (f4f3f1f9f4f1f10) ∪ (f4f3f1f9f4f1f8) ∪ S S (f4f3f1f9f7f4f1f8) ∪ (f4f3f1f4f1f5) ∪ S S S S (f9f2f10) ∪ (f9f2f8) ∪ (f7f2f8) ∪ (f2f5) ∪ S S S (f9f1f10) ∪ (f9f1f8) ∪ (f7f1f8) ∪ S S S (f1f5) ∪ (f4f1f10) ∪ (f4f1f8) ∪ S S (f4f3f1f2f10) ∪ (f4f3f1f2f8) Number of involutions 1+4+4+8+2+8+8+16+2+8+8+16+4+8+8+16

∼ If G = A15 then there are 3 conjugacy classes of involutions in G.

The ﬁrst one consists of 8 conjugacy classes of involutions in S with the following number of involutions: 1+2+2+2+4+4+4+8.

The second one consists of 19 conjugacy classes of involutions in S with the following number of involutions: 1+2+4+4+4+4+4+4+4+4+8+8+8+8+8+8+8+8+16.

The third one consists of 19 conjugacy classes of involutions in S with the following number of involutions: 1+2+2+4+4+4+8+8+8+8+8+8+8+16+16+16.

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