REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 7, Pages 300{321 (July 29, 2003) S 1088-4165(03)00175-4 THE FISCHER-CLIFFORD MATRICES 0 OF A MAXIMAL SUBGROUP OF Fi24 FARYAD ALI AND JAMSHID MOORI Abstract. 0 The Fischer group Fi24 is the largest sporadic simple Fischer group of order 1255205709190661721292800 = 221:316:52:73:11:13:17:23:29 : 0 The group Fi24 is the derived subgroup of the Fischer 3-transposition group Fi24 discovered by Bernd Fischer. There are five classes of elements of order 0 3inFi24 as represented in ATLAS by 3A,3B,3C,3D and 3E. A subgroup 0 2f g of Fi24 of order 3 is called of type 3X,whereX A; B; C; D; E ,ifitisgen- erated by an element in the class 3X. There are six classes of maximal 3-local 0 subgroups of Fi24 as determined by Wilson. In this paper we determine the Fischer-Clifford matrices and conjugacy classes of one of these maximal 3-local ¯ h i ∼ 7· ∼ 7 subgroups G := NFi0 ( N ) = 3 O7(3), where N = 3 is the natural orthog- 24∼ onal module for G=N¯ = O7(3) with 364 subgroups of type 3B corresponding to the totally isotropic points. The group G¯ is a nonsplit extension of N by ∼ G = O7(3). 1. Introduction Let G¯ = N·G be an extension of N by G,whereN is a normal subgroup of G¯ ∼ and G=N¯ = G.Let¯g 2 G¯ be a lifting of g 2 G under the natural homomorphism G¯ −→ G and let [g] be a conjugacy class of elements of G with representative g. Let X(g)=fx1;x2;:::;xc(g)g be a set of representatives of the conjugacy classes of G¯ from the coset Ng¯ whose images under the natural homomorphism G¯ −→ G are in [g]andwetakex1 =¯g.Letfθ1,θ2,...,θtg be a set of representatives of ¯ ≤ ≤ ¯ the orbits of G on Irr(N) such that for 1 i t,wehaveHi = IG¯ (θi)withthe corresponding inertia factors Hi and let i be a projective character of H¯i with the factor setα ¯i such that ( i)N = θi. By Gallagher [18] we have [t ¯ f ¯ G¯ j 2 −1g Irr(G)= ( iβ) β IrrProj(Hi); with the factor set αi ; i=1 where αi is obtained fromα ¯i and β¯ from β. Without loss of generality, suppose that θ1 =1N is the identity character of N.ThenH¯1 = G¯ and H1 = G.Nowchoose −1 y1;y2;:::;yr to be the representatives of the αi -conjugacy classes of elements of 2 ≤ ≤ 2 ¯ Hi that fuse to [g]inG.Sinceyk Hi for 1 k r, then we define ylk Hi such ¯ that ylk ranges over all representatives of the conjugacy classes of elements of Hi Received by the editors August 29, 2002 and, in revised form, April 7, 2003. 2000 Mathematics Subject Classification. Primary 20C15, 20D08, 20E22. The first author was supported by a postgraduate bursary from the NRF(SA). The second author was supported by a research grant from University of Natal and NRF(SA). c 2003 American Mathematical Society 300 FISCHER-CLIFFORD MATRICES OF A MAXIMAL SUBGROUP 301 which map to yk under the homomorphism H¯i −→ Hi whose kernel is N.Nowby using the formula for induced characters, we have X X0 j j ¯ G¯ CG¯ (xj) ¯ ( iβ) (xj )= iβ(y`k ) jC ¯ (y` )j 1≤k≤r ` Hi k X X0 j j CG¯ (xj) ¯ = i(y`k )β(y`k ) jC ¯ (y` )j 1≤k≤r ` Hi k X X0 j j CG¯ (xj ) = ( i(y`k ))β(yk) jC ¯ (y` )j 1≤k≤r ` Hi k P 0 ∼ ¯ where ` is the summation over all ` for which y`k xj in G. Now we define a matrix Mi(g)byMi(g)=(auv), where 1 ≤ u ≤ r and 1 ≤ v ≤ c(g), and X0 j j CG¯ (xj) auv = i(y`k ) : jC ¯ (y` )j ` Hi k Then we get X G¯ ( iβ) (xj )= auvβ^(yk) : 1≤k≤r By doing so for all 1 ≤ i ≤ t such that Hi contains an element in [g] we obtain the matrix M(g)givenby 2 3 M1(g) 6 7 6 M2(g) 7 6 7 M(g)=4 . 5 ; . Mt(g) where Mi(g) is the submatrix corresponding to the inertia group H¯i and its inertia factor Hi.IfHi \ [g]=;,thenMi(g)willnotexistandM(g) does not contain × −1 Mi(g). The size of the matrix M(g)isl c(g)wherel is the number of αi - regular conjugacy classes of elements of the inertia factors Hi for 1 ≤ i ≤ t which fuse into [g]inG,andc(g) is the number of conjugacy classes of elements of G¯ which correspond to the cosetgN ¯ .ThenM(g)istheFischer-Clifford matrix of G¯ corresponding to the cosetgN ¯ . WewillseelaterthatM(g)isac(g) × c(g) nonsingular matrix. Let R(g)=f(i; yk) j 1 ≤ i ≤ t; Hi \ [g] =6 ; ; 1 ≤ k ≤ rg ; −1 and we note that yk runs over representatives of the αi -conjugacy classes of el- ements of Hi which fuse into [g]inG. Following the notation used by Fischer [9], Mpono [25] and Whitely [30] we denote M(g)bywritingM(g)=Cl(Ng¯)= (i;yk) (aj ), where X0 j j (i;yk) CG¯ (xj ) aj = i(y`k ) ; jC ¯ (y` )j ` Hi k with columns indexed by X(g) and rows indexed by R(g). Then the partial char- ¯ acter table of G on the classes fx1;x2;:::;xc(g)g is given by 2 3 C1(g)M1(g) 6 7 6 C2(g)M2(g) 7 6 7 4 . 5 . Ct(g)Mt(g) 302 FARYAD ALI AND JAMSHID MOORI where the Fischer-Clifford matrix M(g) is divided into blocks with each block cor- responding to an inertia group H¯i and Ci(g) is the partial projective character table −1 −1 of Hi with the factor set αi consisting of the columns corresponding to the αi - regular classes that fuse into [g]inG. We obtain the characters of G¯ by multiplying −1 the relevant columns of the projective characters of Hi with the factor set αi by the rows of M(g). We can also observe that the number of irreducible characters of G¯ is the sum of numbers of projective characters of the inertia factors Hi with −1 ≤ ≤ the factor set αi , for all i,1 i t. In the present paper we determine Fischer-Clifford matrices and conjugacy classes 7 7 of 3 · O7(3). Then the character table of 3 · O7(3) can be constructed by using these matrices and the partial character tables of the inertia factor groups. Let K be a group and A ≤ Aut(K). Then by Brauer's theorem (see [14]) A acts on the conjugacy classes of elements of K and on the irreducible characters of K resulting in the same number of orbits. Lemma 1.1. Suppose we have the following matrix describing the above actions: 1=l1 l2 ··· lj ··· lt 0 1 s1 11··· 1 ··· 1 B ··· ··· C s2 B a21 a22 a2j a2t C B C . B . C . B . C . B . C B ··· ··· C si B ai1 ai2 aij ait C B C . @ . A . st at1 at2 ··· atj ··· att where a1j =1for j 2f1; 2;:::;tg, lj's are lengths of orbits of A on the conjugacy classes of K, si's are lengths of orbits of A on Irr(K) and aij is the sum of si irreducible characters of K on the element xj ,wherexj is an element of the orbit 0 2f g Pof length lj. Then the following relation holds for i; i 1; 2;:::;t : t 0 j j 0 j=1 aij ai jlj = K siδii . Proof. This result has been proved as Lemma 2.2.2 in [27] and as Lemma 4.2.2 in [30]. For arithmetical properties the weights are important and hence we attach to M(g) the corresponding weights.Letxj 2 X(g). For a fixed coset X =¯gN 2 G=N¯ , we define mj =[NG¯ (X):CG¯ (xj )]. The Fischer-Clifford matrix M(g) is partitioned row-wise into blocks, where each block corresponds to an inertia group. The columns of M(g) are indexed by X(g) 2 j j and for each xj X(g), at the top of the columns of M(g), we write CG¯ (xj ) and at the bottom we write mj.TherowsofM(g) are indexed by R(g)andontheleft j j of each row we write CHi (yk) ,whereyk fuses into [g]inG. Then, in general, we can write M(g) with corresponding weights for rows and columns as follows, where blocks corresponding to the inertia groups are separated by horizontal lines: FISCHER-CLIFFORD MATRICES OF A MAXIMAL SUBGROUP 303 jC (x )jjC (x )|···jC (x )j 0 G 1 G 2 G c(g) 1 B (1;g) (1;g) (1;g) C jCG(g)j B a a ::: a C B 1 2 c(g) C B C B C B C B (2;y1) (2;y1) (2;y1) C jC (y1)j B a a ::: a C H2 B 1 2 c(g) C B (2;y2) (2;y2) (2;y2) C jC (y2)j B a a ::: a C H2 B 1 2 c(g) C B C . B . C . B . C B C B C B C B C B C jC (y )j B a(i;y1) a(i;y1) ::: a(i;y1) C Hi 1 B 1 2 c(g) C B C jC (y )j B a(i;y2) a(i;y2) ::: a(i;y2) C Hi 2 B 1 2 c(g) C B C .