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Article: Zappa, Emilio, Dykeman, Eric C and Twarock, Reidun orcid.org/0000-0002-1824-2003 (2014) On the subgroup structure of the hyperoctahedral group in six dimensions. Acta crystallographica. Section A, Foundations and advances. pp. 417-428. ISSN 2053-2733 https://doi.org/10.1107/S2053273314007712

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Acta Crystallographica Section A Foundations and On the subgroup structure of the hyperoctahedral Advances group in six dimensions ISSN 2053-2733 Emilio Zappa,a,c* Eric C. Dykemana,b,c and Reidun Twarocka,b,c Received 17 January 2014 Accepted 7 April 2014 aDepartment of Mathematics, University of York, York, UK, bDepartment of Biology, University of York, York, UK, and cYork Centre for Complex Systems Analysis, University of York, York, UK. Correspondence e-mail: [email protected]

The subgroup structure of the hyperoctahedral group in six dimensions is investigated. In particular, the subgroups isomorphic to the icosahedral group are studied. The orthogonal crystallographic representations of the icosahedral group are classified and their intersections and subgroups analysed, using results from graph theory and their spectra.

1. Introduction denoted by B6, which is a subgroup of O 6 and can be represented in the standard basis of R6 as theð setÞ of all 6 6 Â The discovery of quasicrystals in 1984 by Shechtman et al. has orthogonal and integral matrices. The subgroups of B6 which spurred the mathematical and physical community to develop are isomorphic to the icosahedral group constitute the integral mathematical tools in order to study structures with non- representations of it; among them, the crystallographic ones crystallographic symmetry. are those which split, in GL 6; R , into two three-dimensional Quasicrystals are alloys with five-, eight-, ten- and 12-fold irreducible representationsð of Þ. Therefore, they carry two symmetry in their atomic positions (Steurer, 2004), and subspaces in R3 which are invariantI under the action of and therefore they cannot be organized as (periodic) lattices. In can be used to model the quasiperiodic structures. I crystallographic terms, their symmetry group G is noncrys- The embedding of the icosahedral group into B6 has been tallographic. However, the noncrystallographic symmetry used extensively in the crystallographic literature. Katz (1989), leaves a lattice invariant in higher dimensions, providing an Senechal (1995), Kramer & Zeidler (1989), Baake & Grimm integral representation of G. If such a representation is (2013), among others, start from a six-dimensional crystal- reducible and contains a two- or three-dimensional invariant lographic representation of to construct three-dimensional subspace, then it is referred to as a crystallographic repre- Penrose tilings and icosahedralI quasicrystals. Kramer (1987) sentation, following terminology given by Levitov & Rhyner and Indelicato et al. (2011) also apply it to study structural (1988). This is the starting point to construct quasicrystals via transitions in quasicrystals. In particular, Kramer considers in the cut-and-project method described by, among others, B6 a representation of and a representation of the octahe- Senechal (1995), or as a model set (Moody, 2000). dral group which shareI a tetrahedral subgroup, and defines In this paper we are interested in icosahedral symmetry. The a continuousO rotation (called Schur rotation) between cubic icosahedral group consists of all the rotations that leave a and icosahedral symmetry which preserves intermediate regular icosahedronI invariant, it has size 60 and it is the largest tetrahedral symmetry. Indelicato et al. define a transition of the finite subgroups of SO 3 . contains elements of order between two icosahedral lattices as a continuous path five, therefore it is noncrystallographicð Þ I in three dimensions; connecting the two lattice bases keeping some symmetry the (minimal) crystallographic representation of it is six- preserved, described by a maximal subgroup of the icosahe- dimensional (Levitov & Rhyner, 1988). The full icosahedral dral group. The rationale behind this approach is that the two group, denoted by h, also contains the reflections and is equal corresponding lattice groups share a common subgroup. These to C , where CI denotes the cyclic group of order two. two approaches are shown to be related (Indelicato et al., IÂ 2 2 I h is isomorphic to the Coxeter group H3 (Humphreys, 1990) and 2012), hence the idea is that it is possible to study the transi- is made up of 120 elements. In this work, we focus on the tions between icosahedral quasicrystals by considering two icosahedral group I because it plays a central role in appli- distinct crystallographic representations of in B6 which cations in virology (Indelicato et al., 2011). However, our share a common subgroup. I considerations apply equally to the larger group h. These papers motivate the idea of studying in some detail I R6 Levitov & Rhyner (1988) classified the Bravais lattices in the subgroup structure of B6. In particular, we focus on the that are left invariant by : there are, up to equivalence, subgroups isomorphic to the icosahedral group and its exactly three lattices, usuallyI referred to as icosahedral subgroups. Since the group is quite large (it has 266! elements), Bravais lattices, namely the simple cubic (SC), body-centred we use for computations the software GAP (The GAP Group, cubic (BCC) and face-centred cubic (FCC). The point group 2013), which is designed to compute properties of finite of these lattices is the six-dimensional hyperoctahedral group, groups. More precisely, based on Baake (1984), we generate

Acta Cryst. (2014). A70, 417–428 doi:10.1107/S2053273314007712 417 research papers

the elements of B6 in GAP as a subgroup of the symmetric Table 1 Character table of the icosahedral group. group S12 and then find the classes of subgroups isomorphic to the icosahedral group. Among them we isolate, using results Note that  p5 1 =2 is the golden ratio. ¼ð þ Þ from character theory, the class of crystallographic repre- ffiffiffi sentations of . In order to study the subgroup structure of this class, we proposeI a method using graph theory and their spectra. In particular, we treat the class of crystallographic representations of as a graph: we fix a subgroup of and say that two elementsI in the class are adjacent if theirG I inter- section is equal to a subgroup isomorphic to . We call the resulting graph -graph. These graphs are quiteG large and difficult to visualize;G however, by analysing their spectra 1 (Cvetkovic et al., 1995) we can study in some detail their BM B ; à BM MÀ à B M; M GL n; Z : topology, hence describing the intersection and the subgroups Pð Þ¼Pð Þ ð Þ¼ ð Þ 2 ð Þ shared by different representations. We notice that a change of basis in the lattice leaves the The paper is organized as follows. After recalling, in 2, the point group invariant, whereas the corresponding lattice definitions of point group and lattice group, we define,x in 3, groups are conjugated in GL n; Z . Two lattices are inequi- ð Þ the crystallographic representations of the icosahedral groupx valent if the corresponding lattice groups are not conjugated and the icosahedral lattices in six dimensions. We provide, in GL n; Z (Pitteri & Zanzotto, 2002). ð Þ following Kramer & Haase (1989), a method for the As a consequence of the crystallographic restriction [see, construction of the projection into three dimensions using for example, Baake & Grimm (2013)] five- and n-fold tools from the representation theory of finite groups. In 4we symmetries, where n is a natural number greater than six, are classify, with the help of GAP, the crystallographic repre-x forbidden in dimensions two and three, and therefore any sentations of .In 5 we study their subgroup structure, group G containing elements of such orders cannot be the introducing theI conceptx of -graph, where is a subgroup point group of a two- or three-dimensional lattice. We there- of . G G fore call these groups noncrystallographic. In particular, three- I dimensional icosahedral lattices cannot exist. However, a noncrystallographic group leaves some lattices invariant in 2. Lattices and noncrystallographic groups higher dimensions and the smallest such dimension is called b ... Rn R Let i, i 1; ; n be a basis of , and let B GL n; be the minimal embedding dimension. Following Levitov & ¼ 2 ð b Þ the matrix whose columns are the components of i with Rhyner (1988), we introduce: e ; i ; ...; n Rn respect to the canonical basis i 1 of . A lattice Definition 2.1. Let G be a noncrystallographic group. A in Rn is a Z-free module of rankf n¼with basisgB, i.e. crystallographic representation  of G is a D-dimensional n representation of G such that: x b : Z B mi i mi : (1) the characters of  are integers; Lð Þ¼ ¼ i 1 2   ¼  P (2)  is reducible and contains a two- or three-dimensional Any other lattice basis is given by BM, where M GL n; Z , representation of G. the set of invertible matrices with integral entries2 (whoseð Þ We observe that the first condition implies that G must be determinant is equal to 1) (Artin, 1991). the subgroup of the point group of a D-dimensional lattice. The point group of a latticeÆ is given by all the orthogonal The second condition tells us that  contains either a two- transformations that leave theL lattice invariant (Pitteri & or three-dimensional invariant subspace E of RD, usually Zanzotto, 2002): referred to as physical space (Levitov & Rhyner, 1988). B Q O n : M GL n; Z s:t: QB BM : Pð Þ¼f 2 ð Þ 9 2 ð Þ ¼ g 1 We notice that, if Q B , then BÀ QB M GL n; Z . 2Pð Þ ¼ 2 ð Þ In other words, the point group consists of all the orthogonal 3. Six-dimensional icosahedral lattices matrices which can be represented in the basis B as integral The icosahedral group is generated by two elements, g and matrices. The set of all these matrices constitute the lattice 2 g , such that g2 g3 I g g 5 e, where e denotes the group of the lattice: 3 2 ¼ 3 ¼ð 2 3Þ ¼ identity element. It has size 60 and it is isomorphic to A5, the 1 à B M GL n; Z : Q B s:t: M BÀ QB : alternating group of order five (Artin, 1991). Its character ð Þ¼f 2 ð Þ 9 2Pð Þ ¼ g table is given in Table 1. The lattice group provides an integral representation of the From the character table we see that the (minimal) crys- point group and these are related via the equation tallographic representation of is six-dimensional and is given I 6 1 by T T . Therefore, leaves a lattice in R invariant. à B BÀ B B; 1 2 ð Þ¼ Pð Þ LevitovÈ & Rhyner (1988)I proved that the three inequivalent and moreover the following hold (Pitteri & Zanzotto, 2002): Bravais lattices of this type, mentioned in the Introduction and

418 Emilio Zappa et al.  Subgroup structure of the hyperoctahedral group Acta Cryst. (2014). A70, 417–428 research papers

n referred to as icosahedral (Bravais) lattices, are given by, Im Pi is equal to an ni-dimensional subspace Vi of F invar- ð Þ À respectively: iant under i. In our case, we have two projection operators, : R6 R3 x ... : Z Pi , i = 1, 2, corresponding to the IRs T1 and T2, SC x1; ; x6 xi ; ! x 1 xL; ...¼f; x ¼ð: x Z; x Þ x mod22 ;g i; j 1; ...; 6 ; respectively. We assume the image of P1,Im P1 , to be equal to LBCC ¼ ¼ 2 ð 1 6Þ i 2 i ¼ j 8 ¼ e ... ð Þ x 1 ... : Z 6 Ek, and Im P2 E?.If j; j 1; ; 6 is the canonical basis FCC 2 x1; ; x6 xi ; i 1 xi 0 mod2 : 6 ð Þ¼ f ¼ g L È ¼ ¼ ð Þ 2 ¼ ¼ É of R , then a basis of Ek (respectively E?) can be found ee^ : e ... We note thatÈ a basis of the SC latticeP is the canonicalÉ basis considering the set j Pi j; j 1; ; 6 for i 1 6 f ¼ ¼ g ¼ of R . Its point group is given by (respectively i 2) and then extracting a basis i from it. Since dim E =dim¼ E 3, we obtain ee^ ; ee^ B; ee^ , for i Q O 6 : Q M GL 6; Z O 6 GL 6; Z O 6; Z ; k ? i i;1 i;2 i;3 PSC ¼f 2 ð Þ ¼ 2 ð Þg ¼ ð Þ\ ð Þ’ ð Þ = 1, 2. The matrix R can¼ be thus writtenB ¼f as g 1 ð Þ which is the hyperoctahedral group in dimension six. In the R ee^ ; ee^ ; ee^ ; ee^ ; ee^ ; ee^ : 4 ¼ 1;1 1;2 1;3 2;1 2;2 2;3 ð Þ following, we will denote this group by B6, following  basis of E basis of E  Humphreys (1996). We point out that this notation comes k ? from Lie theory: indeed, B6 represents the root system of the Denoting by k and|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}? the 3 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}6 matrices which represent Lie algebra so 13 (Fulton & Harris, 1991). However, the P and P in the bases and  , respectively, we have, by ð Þ 1 2 B1 B2 corresponding reflection group W B6 is isomorphic to the linear algebra hyperoctahedral group in six dimensionsð Þ (Humphreys, 1990). All three lattices have point group B , whereas their lattice 1 k 6 RÀ : 5 groups are different and, indeed, they are not conjugate in ¼ ? ! ð Þ GL 6; Z (Levitov & Rhyner, 1988). ð Þ 1 1 B Since RÀ 0RÀ [cf. equation (2)], we obtain Let be a subgroup of 6 isomorphic to . provides a H¼H (faithful)H integral and orthogonal representationI H of . More- k g v T k v ;? g v T ? v ; 6 R I 1 2 over, if T1 T2 in GL 6; , then is also crystal- ðHð Þ Þ¼ ð ð ÞÞ ðHð Þ Þ¼ ð ð ÞÞ ð Þ H’ È ð Þ H 6 lographic (in the sense of Definition 2.1). All of the other for all g and v R . In particular, the following diagram 1 2I 2 crystallographic representations are given by BÀ B, where commutes B GL 6; R is a basis of an icosahedral latticeH in R6. Therefore2 ð weÞ can focus our attention, without loss of gener- ality, on the orthogonal crystallographic representations.

3.1. Projection operators Let be a crystallographic representation of the icosahe- The set ; is the starting point for the construction of dral group.H splits into two three-dimensional irreducible k quasicrystalsðHvia Þthe cut-and-project method (Senechal, 1995; representationsH (IRs), T and T ,inGL 6; R . This means that 1 2 Indelicato et al., 2012). there exists a matrix R GL 6; R suchð thatÞ 2 ð Þ

1 T1 0 0 : RÀ R : 2 4. Crystallographic representations of H ¼ H ¼ 0 T2 ! ð Þ I From the previous section it follows that the six-dimensional

The two IRs T1 and T2 leave two three-dimensional hyperoctahedral group B6 contains all the (minimal) ortho- subspaces invariant, which are usually referred to as the gonal crystallographic representations of the icosahedral physical (or parallel) space Ek and the orthogonal space E? group. In this section we classify them, with the help of the (Katz, 1989). In order to find the matrix R (which is not unique computer software programme GAP (The GAP Group, 2013). in general), we follow (Kramer & Haase, 1989) and use results from the representation theory of finite groups (for proofs and 4.1. Representations of the hyperoctahedral group B6 further results see, for example, Fulton & Harris, 1991). In Permutation representations of the n-dimensional hyper- particular, let À:G GL n; F be an n-dimensional repre- octahedral group B in terms of elements of S , the symmetric ! ð Þ n 2n sentation of a finite group G over a field F (F = R, C). group of order 2n, have been described by Baake (1984). In By Maschke’s theorem, À splits, in GL n; F ,as this subsection, we review these results because they allow us À ... À À : ð Þ m1 1 mr r, where i G GL ni; F is an ni- to generate B in GAP and further study its subgroup struc- È È ! ð Þ 6 dimensional IR of G. Then the projection operator ture. : n ni Pi F F is given by It follows from equation (1) that B consists of all the ! 6 orthogonal integral matrices. A matrix A a of this kind : ni À ij Pi Àà g g ; 3 T ¼ð Þ ¼ G i ð Þ ð Þ ð Þ must satisfy AA I6, the identity matrix of order six, and g ¼ j j X2I have integral entries only. It is easy to see that these conditions where denotes the complex conjugate of the character of imply that A has entries in {0, 1} and each row and column ÃÀi the representation À . This operator is such that its image contains 1 or 1 only once. TheseÆ matrices are called signed i À

Acta Cryst. (2014). A70, 417–428 Emilio Zappa et al.  Subgroup structure of the hyperoctahedral group 419 research papers permutation matrices. It is straightforward to see that any A B6 can be written in the form NQ, where Q 2is a 6 6 permutation matrix and N is a diagonal matrix with each diagonal entry being either 1 or 1. We can thus À a associate with each matrix in B6 a pair ; , where a Z6 is a vector given by the diagonalð Þ 2 2 elements of N, and  S6 is the permutation associated with Q. The2 set of all these pairs constitutes a group (called the wreath product Z Z of 2 and S6, and denoted by 2 S6; Humphreys, 1996) with the multiplication ruleo given by a; b; : a b; ; ð Þð Þ ¼ð  þ2 Þ Figure 1 A planar representation of an icosahedral surface, showing our labelling convention for the where 2 denotes addition modulo 2 and þ vertices; the dots represent the locations of the symmetry axes corresponding to the a : a ...  k a k ; a1; ; a6 : generators of the icosahedral group and its subgroups. The kite highlighted is a fundamental ð Þ ¼ ð Þ ¼ð Þ domain of the icosahedral group. Z 2 S6 and B6 are isomorphic, an isomorphism o ^ : T being the following: Using equation (11) we obtain a representation B6 given by II I! a : ai T ; ij 1  i ;j: 8 ½ ð ފ ¼ðÀ Þ ð Þ ð Þ 000001 000001 6 It immediately follows that |B6|=26! = 46 080. A set of 000010 000100 0 1 0 1 generators is given by ^ 001000 ^ 0 10000 g À ; g À : IIð 2Þ¼B 000 100C IIð 3Þ¼B 001000C : 0; 1; 2 ; : 0; 1; 2; 3; 4; 5; 6 ; : 0; 0; 0; 0; 0; 1 ; id ; B À C B À C S6 B 010000C B 100000C ¼ð ð ÞÞ ¼ð ð ÞÞ ¼ðð Þ Þ B C B C 9 B 100000C B 000010C ð Þ B C B C @ A @ 12A which satisfy the relations ð Þ

2 2 6 We see that ^ g2 2 and ^ g3 0, so that, by looking 0; id : II ð Þ¼À II ð Þ¼ ¼ ¼ ¼ð S6 Þ at the character table of , we have I : Z Finally, the function ’ 2 S6 S12 defined by ^ ; T1 T2 o ! II ¼ þ a :  k 6ak if 1 k 6 which implies, using Maschke’s theorem (Fulton & Harris, ’ ; k ð Þþ   ^ R ð Þð Þ ¼  k 6 6 1 ak 6 if 7 k 12 1991), that T1 T2 in GL 6; . Therefore, the subgroup ( ð À Þþ ð À À Þ   ^ of B is aII’ crystallographicÈ representationð Þ of . 10 II 6 I ð Þ Before we continue, we recall the following (Humphreys, is injective and maps any element of Z S into a permutation 1996): 2 o 6 of S12, and provides a faithful permutation representation of B6 as a subgroup of S12. Combining equation (8) with the Definition 4.1. Let H be a subgroup of a group G.The inverse of equation (10) we get the function conjugacy class of H in G is the set 1 : 1 : : T À : S B ; 11 G H gHgÀ g G : ¼  12 ! 6 ð Þ C ð Þ ¼f 2 g which can be used to map a permutation into an element of B6. In order to find all the other crystallographic representa- 4.2. Classification tions, we use the following scheme:

In this subsection we classify the orthogonal crystal- (a) we generate B6 as a subgroup of S12 using equations (9) lographic representations of the icosahedral group. We start and (10); by recalling a standard way to construct such a representation, (b) we list all the conjugacy classes of the subgroups of B6 following Zappa et al. (2013). We consider a regular icosahe- and find a representative for each class; dron and we label each vertex by a number from one to 12, so (c) we isolate the classes whose representatives have order that the vertex opposite to vertex i is labelled by i 6 (see 60; þ Fig. 1). This labelling induces a permutation representation (d) we check if these representatives are isomorphic to ;  : S given by (e) we map these subgroups of S into B using equationI I! 12 12 6 (11) and isolate the crystallographic ones by checking the  g 1; 6 2; 5 3; 9 4; 10 7; 12 8; 11 ; 2 characters; denoting by S the representative, we decompose ð g Þ¼ð1; 5Þð; 6 2Þð; 9; 4Þð7; 11Þð; 12 3Þð; 10; 8 Þ: S ð 3Þ¼ð Þð Þð Þð Þ as

420 Emilio Zappa et al.  Subgroup structure of the hyperoctahedral group Acta Cryst. (2014). A70, 417–428 research papers

S m1 A m2 T m3 T m4 G m5 H ; B ¼ þ 1 þ 2 þ þ able fact is that they split into two different classes in 6þ.To N ... mi ; i 1; ; 5: see this, we first need to generate B6þ. In particular, with GAP 2 ¼ we isolate the subgroups of index two in B , which are normal Note that S is crystallographic if and only if m m 1 6 2 3 in B , and then, using equation (11), we find the one whose and m m m 0. ¼ ¼ 6 1 4 5 generators have determinant equal to one. In particular, we We implemented¼ ¼ ¼ steps (1)–(4) in GAP (see Appendix C). have There are three conjugacy classes of subgroups isomorphic to in B . Denoting by S g ; g the representatives of the Bþ 1; 2; 6; 4; 3 7; 8; 12; 10; 9 ; 5; 11 6; 12 ; I 6 i ¼h 2;i 3;ii 6 ¼hð Þð Þ ð Þð Þ classes returned by GAP, we have, using equation (11), 1; 2; 6; 5; 3 7; 8; 12; 11; 9 ; 5; 12; 11; 6 : ð Þð Þ ð Þi g 2; g 3 2 S 2A G; S1 ð 2;1Þ¼ S1 ð 3;1Þ¼ ) S1 ¼ A þ G ) 1 ’ È We can then apply the same procedure to find the crystal- g 2; g 0 S T T ; lographic representations of , and see that they split into two S2 ð 2;2Þ¼À S2 ð 3;2Þ¼ ) S2 ¼ T1 þ T2 ) 2 ’ 1 È 2 classes, each one of size 96.I Again we can choose ^ as a S g2;3 2; S g3;3 0 S A H S3 A H: 3 ð Þ¼ 3 ð Þ¼ ) 3 ¼ þ ) ’ È representative for one of these classes; a representativeII ^ for Since 2A is decomposable into two one-dimensional the other one is given by KK representations, it is not strictly speaking two dimensional in the sense of Definition 2.1, and as a consequence, only the 010000 000100 100000 100000 second class contains the crystallographic representations of 0 1 0 1 ^ 00 1000 00000 1 . A computation in GAP shows that its size is 192. We thus À ; À : I KK¼ B 000001C B 010000C have the following: *B C B C+ B 0000 10C B 00 1000C B À C B À C Proposition 4.1. The crystallographic representations of in B 000100C B 000010C B I B C B C 6 form a unique conjugacy class in the set of all the classes of @ A @ A13 subgroups of B6, and its size is equal to 192. ð Þ We note that in the more general case of h,wecan We briefly point out that the other two classes of subgroups construct the crystallographic representations ofI starting isomorphic to in B have an interesting algebraic intepre- I h I 6 from the crystallographic representations of . First of all, we tation. First of all, we observe that B6 is an extension of S6, recall that I C , where C is the cyclicI group of order since according to Humphreys (1996): I h ¼  2 2 two. Let be a crystallographic representation of in B6, and Z6 Z Z6 À H I B6= 2 2 S6 = 2 S6: let 1; 1 be a one-dimensional representation of C2. ’ð o Þ ’ Then¼f the representationÀ g ^ given by Following Janusz & Rotman (1982), it is possible to embed HH ^ the symmetric group S into S in two different ways. The : À; 5 6 HH ¼H canonical embedding is achieved by fixing a point in 1; ...; 6 where denotes the tensor product of matrices, is a repre- and permuting the other five, whereas the other embeddingf isg sentation of in B and it is crystallographic in the sense of by means of the so-called ‘exotic map’ ’ : S S , which acts I h 6 5 ! 6 Definition 2.1 (Fulton & Harris, 1991). on the six 5-Sylow subgroups of S5 by conjugation. Recalling that the icosahedral group is isomorphic to the alternating 4.3. Projection into the three-dimensional space group A5, which is a normal subgroup of S5, then the canonical embedding corresponds to the representation 2A G in B , We study in detail the projection into the physical space Ek È 6 using the methods described in 3.1. while the exotic one corresponds to the representation A H. ^ x In what follows, we will consider the subgroup ^ previouslyÈ Let be the crystallographic representation of given in II equationII (12). Using equation (3) with n 3 andI G defined as a representative of the class of the crystallographic i ¼ j j¼ representations of , and denote this class by ^ . 60 we obtain the following projection operators B6 jIj ¼ I 1 C 2ðIIÞ Recalling that two representations Dð Þ and Dð Þ of a group p5 1 1 111 G are said to be equivalent if there are related via a similarity 1 p5 À1 À1 11 0 ffiffiffi 1 transformation, i.e. there exists an invertible matrix S such that 1 11p5 À1 À11 P1 À ffiffiffi À ; 1 2 1 ¼ 2p5B 1 11p5 11C Dð Þ SDð ÞSÀ ; B À À ffiffiffi C ¼ B 1 1 11p5 1 C B ffiffiffi C then an immediate consequence of Proposition 4.1 is the ffiffiffiB 11111À À p5 C B ffiffiffi C following: @ A ffiffiffi Corollary 4.1. Let 1 and 2 be two orthogonal crystal- lographic representationsH of H.Then and are equiva- p5 1111 1 1 2 À À À lent in B . I H H 1 p5 1111 6 0 Àffiffiffi À À 1 ^ 1 1 1 p5 111 We observe that the determinant of the generators of in P2 Àffiffiffi À À : ^ II ¼ 2p5B 111 p5 1 1 C equation (12) is equal to one, so that Bþ : B Àffiffiffi À À C 6 B 1111 p5 1 C A B : detA 1 . Proposition 4.1 implies thatII2 all the¼ B ffiffiffi C 6 ffiffiffiB À1 1 1 À1 1 pÀ5 C crystallographicf 2 ¼ representationsg belong to B . The remark- B ffiffiffi C 6þ @ À À À À À A ffiffiffi

Acta Cryst. (2014). A70, 417–428 Emilio Zappa et al.  Subgroup structure of the hyperoctahedral group 421 research papers

Table 2 Table 4 ^ Explicit forms of the IRs T1 and T2 with T1 T2. Permutation representations of the generators of the subgroups of the II’ È icosahedral group. Generator Irrep T1 Irrep T2  g2 1; 6 2; 5 3; 9 4; 10 7; 12 8; 11 ðg Þ¼ð1; 12Þð 2;Þð8 3;Þð4 5; 11Þð 6; 7Þð 9; 10Þ ð 2dÞ¼ð Þð Þð Þð Þð Þð Þ  11   1  1  g3 1; 5; 6 2; 9; 4 7; 11; 12 3; 10; 8 g 1 À 1 À À À ð Þ¼ð Þð Þð Þð Þ 2 1 1  1  1  g3d 1; 10; 2 3; 5; 12 4; 8; 7 6; 9; 11 2 À À 2 À À À ð Þ¼ð Þð Þð Þð Þ 0 1 1 1 0 1  1  1  g5 1; 2; 3; 4; 5 7; 8; 9; 10; 11 À À À À À ðg Þ¼ð1; 10; 11; 3Þð; 6 4; 5; 9; 12;Þ7 @ A @ A ð 5dÞ¼ð Þð Þ

11 11  Table 5 g3 1 À 1 À À À 1  1   1  1 Sizes of the classes of subgroups of the icosahedral group in and B6. 2 2 0 À1 À 1  1 0 À 11À  1 I À À À À Subgroup B @ A @ A jCI ðGÞj jC 6 ðKGÞj 5 480 T 10 6 576 D 10 960 D6 Table 3 4 5 120 D Nontrivial subgroups of the icosahedral group. C5 6 576 C3 10 320 stands for the tetrahedral group, 2n for the dihedral group of size 2n, and C 15 180 T D 2 Cn for the cyclic group of size n. Subgroup Generators Relations Size dimensions centred at the origin (Senechal, 1995; Katz, 1989; g ; g g2 g3 g g 3 e 12 T 2 3d 2 ¼ 3d ¼ð 2 3dÞ ¼ g ; g g2 g5 g g 2 e 10 Indelicato et al., 2011). D10 2d 5d 2d ¼ 5d ¼ð 5d 2dÞ ¼ g ; g g2 g3 g g 2 e Let be another crystallographic representation of in B6. 6 2d 3 2d 3 3 2d 6 K ^ I D 5 ¼ ¼ð Þ ¼ By Proposition 4.1, and are conjugated in B . Consider C5 g5d g5d e 5 6 2 ¼ 2 2 K^ 1 II 4 g2d; g2 g2d g2 g2g2d e 4 M B6 such that M MÀ and let S MR. We have D 3 ¼ ¼ð Þ ¼ 2 II ¼K ¼ C3 g3 g3 e 3 1 1 1 1 1 ^ ¼ S S MR À MR R M MR R R T T : C g g2 e 2 À À À À 1 2 2 2 2 ¼ K ¼ð Þ Kð Þ¼ K ¼ II ¼ È Therefore it is possible, with a suitable choice of the reducing matrices, to project all the crystallographic representations of

The rank of these operators is equal to three. We choose as a in B6 in the same physical space. I basis of Ek and E? the following linear combination of the columns c of the projection operators P , for i = 1, 2 and i;j i 5. Subgroup structure j 1; ...; 6: ¼ The nontrivial subgroups of are listed in Table 3, together c c c c c c c c c c c c with their generators (Hoyle,I 2004). Note that , and 1;1 þ 1;5 ; 1;2 À 1;4 ; 1;3 þ 1;6; 2;1 À 2;5 ; 2;2 þ 2;4 ; 2;3 À 2;6 : T D10 D6 2 2 2 2 2 2 are maximal subgroups of , and that , C and C are ! I D4 5 3 basis of Ek basis of E? normal subgroups of , 10 and 6, respectively (Humphreys, 1996; Artin, 1991). TheT D permutationD representations of the |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}With a suitable rescaling, we obtain|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} the matrix R given by generators in S12 are given in Table 4 (see also Fig. 1). Since is a small group, its subgroup structure can be easily  10 01 I 0  1 1  0 obtained in GAP by computing explicitly all its conjugacy 1 0 10 À0 1  1 classes of subgroups. In particular, there are seven classes of R À À : nontrivial subgroups in : any subgroup H of has the ¼ 2 2  B 0  11 0 C B À C property that, if K is anotherI subgroup of isomorphicI to H, ð þ Þ B  10 01C B À À C then H and K are conjugate in (this propertyI is referred to as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 10 0 1  C B À À C the ‘friendliness’ of the subgroupI H; Soicher, 2006). In other @ A The matrix R is orthogonal and reduces ^ as in equation (2). words, denoting by n the number of subgroups of II isomorphic to , i.e. G I In Table 2 we give the explicit forms of the reduced repre- G sentation. The matrix representation in Ek of P1 is given by n : H< : H ; 14 [see equation (5)] G ¼jf I ’ Ggj ð Þ we have (cf. Definition 4.1)  0 10 1 1 À n : k 1  0  10: G ¼jCI ðGÞj ¼ 2 2  0 01 À10À  1 In Table 5 we list the size of each class of subgroups in . ð þ Þ I @ A Geometrically, different copies of C2, C3 and C5 correspond to pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie e ... The orbit T1 k j , where j; j 1; ; 6 is the cano- the different two-, three- and fivefold axes of the icosahedron, nical basis off Rð6, representsð ÞÞg a regularf ¼ icosahedrong in three respectively. In particular, different copies of stabilize one D10

422 Emilio Zappa et al.  Subgroup structure of the hyperoctahedral group Acta Cryst. (2014). A70, 417–428 research papers

of the six fivefold axes of the icosahedron, and each copy of 6 crystallographic representations of may share subgroups. In stabilizes one of the ten threefold axes. Moreover, it is possibleD order to describe more precisely theI subgroup structure of to inscribe five tetrahedra into a dodecahedron, and each ^ we will use some basic results from graph theory and B6 different copy of the tetrahedral group in stabilizes one of theirC ðIIÞ spectra, which we are going to recall in the next section. these tetrahedra. I 5.2. Some basic results of graph theory and their spectra 5.1. Subgroups of the crystallographic representations of I In this section we recall, without proofs, some concepts and Let be a subgroup of . The function (11) provides a results from graph theory and spectral graph theory. Proofs G I representation of in B6, denoted by , which is a subgroup and further results can be found, for example, in Foulds (1992) of ^. Let us denoteG by the conjugacyKG class of in B . and Cvetkovic et al. (1995). B6 6 TheII next lemma showsC ðK thatGÞ this class containsK allG the Let G be a graph with vertex set V v ; ...; v .The ¼f 1 ng subgroups of the crystallographic representations of in B6. number of edges incident with a vertex v is called the degree of I v. If all vertices have the same degree d, then the graph is ^ Lemma 5.1. Let be a crystallographic repre- called regular of degree d. A walk of length l is a sequence of l Hi 2CB6 ðIIÞ sentation of in B and let be a subgroup of consecutive edges and it is called a path if they are all distinct. I 6 Ki Hi Hi isomorphic to . Then i B . A circuit is a path starting and ending at the same vertex and G K 2C 6 ðKGÞ the girth of the graph is the length of the shortest circuit. Two Proof. Since C ^ , there exists g B such that vertices p and q are connected if there exists a path containing i B6 6 1 ^ H 2 ðIIÞ 1 2 ^ p and q. The connected component of a vertex v is the set of all g igÀ , and therefore g igÀ 0 is a subgroup of isomorphicH ¼ II to . Since all theseK subgroups¼K are conjugate inII vertices connected to v. ^ G [they are ‘friendly’ in the sense intended by Soicher The adjacency matrix A of G is the n n matrix A aij II ^ 1  ¼ð Þ (2006)], there exists h such that h 0hÀ . Thus whose entries aij are equal to one if the vertex vi is adjacent to 1 G 2 II K ¼K & the vertex v , and zero otherwise. It can be seen immediately hg i hg À , implying that i B . j ð ÞK ð Þ ¼KG K 2C 6 ðKGÞ from its definition that A is symmetric and aii 0 for all i,so that Tr A 0. It follows that A is diagonalisable¼ and all its We next show that every element of is a subgroup ð Þ¼ B6 eigenvalues are real. The spectrum of the graph is the set of all of a crystallographic representation of C. ðKGÞ I the eigenvalues of its adjacency matrix A, usually denoted by ^  A . Lemma 5.2.Let i B . There exists i B ð Þ K 2C 6 ðKGÞ H 2C 6 ðIIÞ such that i is a subgroup of i. Theorem 5.1. Let A be the adjacency matrix of a graph G K H with vertex set V v ; ...; v . Let N i; j denote the ¼f 1 ng kð Þ Proof.Since , there exists g B such that number of walks of length k starting at vertex vi and finishing i B6 6 1 K 2C ðKGÞ: 1 ^ 2 at vertex v . We have g igÀ . We define i gÀ g. It can be seen imme- j G K ¼K H ¼ II & diately that i is a subgroup of i. N i; j Ak: K H kð Þ¼ ij As a consequence of these lemmata, contains all We recall that the spectral radius of a matrix A is defined by B6 the subgroups of B which are isomorphicC ðKG toÞ and are  A : max  :   A .IfA is a non-negative matrix, i.e. 6 ð Þ ¼ fj j 2 ð Þg subgroups of a crystallographic representationG of .The if all its entries are non-negative, then  A  A (Horn & ð Þ2 ð Þ explicit forms of are given in Appendix B. We pointI out Johnson, 1985). Since the adjacency matrix of a graph is non- that it is possibleK toG find subgroups of B isomorphic to a negative,   A : r, where   A and r is the largest 6 j j ð Þ ¼ 2 ð Þ subgroup of which are not subgroups of any crystal- eigenvalue. r is called the index of the graph G. lographicG representationI of . For example, the following I ... subgroup Theorem 5.2. Let 1; ;n be the spectrum of a graph G, and let r denote its index. Then G is regular of degree r if and 100000 001000 only if 0 10000 100000À 0 À 1 0 1 n ^ 001000 0 10000 1 2 À ; À i r: TT¼ B 000100C B 000001C n i 1 ¼ *B À C B C+ ¼ B 000010C B 000100C X B À C B C Moreover, if G is regular the multiplicity of its index is equal B 000001C B 000010C B C B C to the number of its connected components. @ A @ A is isomorphic to the tetrahedral group ; a computation in GAP shows that it is not a subgroup of anyT elements in ^ . B6 Indeed, the two classes of subgroups, and C^ ,ð areIIÞ 5.3. Applications to the subgroup structure B6 B6 disjoint. C ðKT Þ C ðTTÞ Let be a subgroup of . In the following we represent the G I Using GAP, we compute the size of each B (see Table subgroup structure of the class of crystallographic repre- ^ C 6 ðKGÞ ^ 5). We observe that B < B n . This implies that sentations of in B6, B , as a graph. We say that jC 6 ðKGÞj jC 6 ðIIÞjÁ G I C 6 ðIIÞ

Acta Cryst. (2014). A70, 417–428 Emilio Zappa et al.  Subgroup structure of the hyperoctahedral group 423 research papers

; ^ are adjacent to each other (i.e. connected by We notice that, using Theorem 5.2, not all the graphs are H1 H2 2CB6 ðIIÞ an edge) in the graph if there exists P B such that connected. In particular, the 10- and the 6-graphs are made 2C 6 ðKGÞ D D P 1 2. We can therefore consider the graph up of six connected components, whereas the C3- and the C2- G ¼H \H^ ; E , where an edge e E is of the form ; . graphs consist of two connected components. With GAP,we B6 1 2 We¼ðC call thisðIIÞ graphÞ -graph. 2 ðH H Þ implemented a breadth-first search algorithm (Foulds, 1992), Using GAP, weG compute the adjacency matrices of the which starts from a vertex i and then ‘scans’ for all the vertices -graphs. The algorithms used are shown in Appendix C. The connected to it, which allows us to find the connected spectraG of the -graphs are given in Table 6. We first of all components of a given -graph (see Appendix C). We find notice that theG adjacency matrix of the C -graph is the null that each connected componentG of the -and -graphs is 5 D10 D6 matrix, implying that there are no two representations sharing made up of 32 vertices, while for the C3- and C2-graphs each precisely a subgroup isomorphic to C5, i.e. not a subgroup component consists of 96 vertices. For all other subgroups, the containing C5. We point out that, since the adjacency matrix of corresponding -graph is connected and the connected the -graph is not the null one, then there exist crystal- component containsG trivially 192 vertices. D10 lographic representations, say i and j, sharing a maximal We now consider in more detail the case when is a subgroup isomorphic to . SinceH C isH a (normal) subgroup maximal subgroup of . Let ^ and let us considerG its D10 5 I H2CB6 ðIIÞ of 10, then i and j do share a C5 subgroup, but also a C2 vertex star in the corresponding -graph, i.e. subgroup.D InH otherH words, if two representations share a G fivefold axis, then necessarily they also share a twofold axis. V : ^ : is adjacent to : 15 B6 A straightforward calculation based on Theorem 5.2 leads ðHÞ ¼fK2C ðIIÞ K Hg ð Þ to the following A comparison of Tables 5 and 6 shows that d n [i.e. the number of subgroups isomorphic to in , cf. Gequation¼ G (14)] Proposition 5.1. Let be a subgroup of . Then the G I corresponding -graph isG regular. I and therefore, since the graph is regular, V d n . G This suggests that there is a one-to-onej ðHÞj correspondence ¼ G ¼ G between elements of the vertex star of and subgroups of In particular, the degree d of each -graph is equal to the H H G G isomorphic to ; in other words, if we fix any subgroup P of largest eigenvalue of the corresponding spectrum. As a G H consequence we have the following: isomorphic to , then P ‘connects’ with exactly another representation G . We thus have the following:H Proposition 5.2. Let be a crystallographic representation K H of in B6. Then there are exactly d representations Proposition 5.3. Let be a maximal subgroup of . Then for I ^ such that G G I j B6 every P B there exist exactly two crystallographic K 2C ðIIÞ 2C 6 ðKGÞ ^ ... representations of , ; , such that P . j Pj; Pj ; j 1; ; d : 1 2 B6 1 2 H\K ¼ 9 2CðKGÞ ¼ G I H H 2C ðIIÞ ¼H \H In particular, we have d = 5, 6, 10, 0, 30, 20, 60 and 60 for In order to prove it, we first need the following lemma: ; ; , C ; ; CG ; and e , respectively. G¼T D10 D6 5 D4 3 C2 f g Lemma 5.3. Let be a maximal subgroup of . Then the In particular, this means that for any crystallographic corresponding -graphG is triangle-free, i.e. it has noI circuits of representation of there are precisely d other such repre- length three. G sentations which shareI a subgroup isomorphicG to . In other words, we can associate to the class ^ theG ‘subgroup Proof. Let A be the adjacency matrix of the -graph. By B6 matrix’ S whose entries are defined by C ðIIÞ Theorem 5.1, itsG third power A3 determines theG number of walks of length three, and in particularG its diagonal entries, S ; i; j 1; ...; 192: ij i j 3 ... ¼jH \Hj ¼ A ii, for i 1; ; 192, correspond to the number of trian- ð GÞ ¼ The matrix S is symmetric and Sii 60, for all i, since the gular circuits starting and ending in vertex i.Adirect ¼ 3 order of is 60. It follows from Proposition 5.2 that each row computation shows that A ii 0, for all i, thus implying the I ð GÞ ¼ of S contains d entries equal to . Moreover, a rearrange- non-existence of triangular circuits in the graph. & ment of the columnsG of S shows thatjGj the 192 crystallographic representations of can be grouped into 12 sets of 16 such that any two of theseI representations in such a set of 16 share a Proof of Proposition 5.3.IfP , then, using Lemma B6 -subgroup. This implies that the corresponding subgraph of 5.2, there exists ^ such2C thatðKPGÞis a subgroup of . D4 H1 2CB6 ðIIÞ H1 the 4-graph is a complete graph, i.e. every two distinct Let us consider the vertex star V 1 . We have V 1 d ; D ... ðH Þ j ðH Þj ¼ G vertices are connected by an edge. From a geometric point of we call its elements 2; ; d 1. Let us suppose that P is H H Gþ ... view, these 16 representations correspond to ‘six-dimensional not a subgroup of any j, for j 2; ; d 1. This implies icosahedra’. This ensemble of 16 such icosahedra embedded that P does not connectH with¼ any of theseG þ . However, H1 Hj into a six-dimensional hypercube can be viewed as a six- since 1 has exactly n different subgroups isomorphic to , H G G dimensional analogue of the three-dimensional ensemble of then at least two vertices in the vertex star, say 2 and 3, are five tetrahedra inscribed into a dodecahedron, sharing pair- connected by the same subgroup isomorphic toH , whichH we G wise a C3-subgroup. denote by Q. Therefore we have

424 Emilio Zappa et al.  Subgroup structure of the hyperoctahedral group Acta Cryst. (2014). A70, 417–428 research papers

Q 1 2; Q 1 3 Q 2 3: Table 6 ¼H \H ¼H \H ) ¼H \H Spectra of the -graphs for a nontrivial subgroup of and e ,the G G I G¼f g This implies that , and form a triangular circuit in the trivial subgroup consisting of only the identity element e. H1 H2 H3 graph, which is a contradiction due to Lemma 5.3, hence the The numbers highlighted are the indices of the graphs, and correspond to their & degrees d . result is proved. G -graph -graph -graph C -graph T D10 D6 5 Eig. Mult. Eig. Mult. Eig. Mult. Eig. Mult. It is noteworthy that the situation in B6þ is different. If we denote by X1 and X2 the two disjoint classes of crystal- 5 1 6 6 10 6 0 192 345 290 290 lographic representations of in B6þ [cf. equation (13)], we I 345 290 290 can build, in the same way as described before, the -graphs À150 À66 À10 6 G À À for X1 and X2, for ; 10 and 6. The result is that the 150 G¼T D D À51 adjacency matrices of all these six graphs are the null matrix of À dimension 96. This implies that these graphs have no edges, and so the representations in each class do not share any -graph C -graph C -graph e -graph D4 3 2 f g maximal subgroup of . As a consequence, we have the Eig. Mult. Eig. Mult. Eig. Mult. Eig. Mult. following: I 30 1 20 2 60 2 60 1 Proposition 5.4. Let ; ^ be two crystallographic 18 5 4 90 4 90 12 5 H K2CB6 ðIIÞ 12 5 4 100 490 490 representations of , and P , P , where is À À B6 615 12 10 490 I ¼H\K 2C ðKGÞ G a maximal subgroup of . Then and are not conjugated in 245 À À12 5 I H K À Bþ. In other words, the elements of B which conjugate with 031 60 1 6 6 H 230 À are matrices with determinant equal to 1. À K À 445 À815 We conclude by showing a computational method which À combines the result of Propositions 4.1 and 5.2. We first recall the following: 000010 000010 À À Definition 5.1. Let H be a subgroup of a group G.The 000100 001000 0 1 0 À 1 normaliser of H in G is given by 001000 000001 0 À ; : H ¼ *B 010000C B 100000C+ 1 B C B C N H : g G : gHgÀ H : B 100000C B 000100C G BÀ C B À C ð Þ ¼f 2 ¼ g B 000001 C B 0 10000C B À C B À C @ A @ A A direct computation shows that the matrix Corollary 5.1. Let and be two crystallographic repre- H K 100000 sentations of in B6 and P such that P . Let I 2CðKGÞ ¼H\K 0 10000 0 À 1 : 1 001000 A ; M B6 M MÀ M H K ¼f 2 H ¼Kg ¼ B 000100C B C be the set of all the elements of B which conjugate with B 000010C 6 B C and let N P be the normaliser of P in B .WehaveH K B 000001C B6 6 B C ð Þ @ A belongs to N and conjugate ^ with . Note that A ; NB H : B6 10 0 H K \ 6 ð Þ 6¼; det M 1. ðKD Þ II H ¼À In other words, it is possible to find a nontrivial element M B in the normaliser of P in B which conjugates with 6 6 6. Conclusions . 2 H K In this work we explored the subgroup structure of the hyperoctahedral group in six dimensions. In particular we Proof. Let us suppose A N H . Then ; B6 1 H K \ ð Þ¼; found the class of the crystallographic representations of the MPMÀ P, for all M A ; . This implies, since 16¼ 2 H K icosahedral group, whose size is 192. Any such representation, M MÀ , that P is not a subgroup of , which is a H ¼K K together with its corresponding projection operator k, can be contradiction. & chosen to construct icosahedral quasicrystals via the cut-and- project method. We then studied in detail the subgroup We give now an explicit example. We consider the repre- structure of this class. For this, we proposed a method based on sentation ^ as in equation (12), and its subgroup (the spectral graph theory and introduced the concept of -graph, 10 explicit formII is given in Appendix B). With GAP, weKD find the for a subgroup of the icosahedral group. This allowedG us to other representation ^ such that ^ . Its study the intersectionG and the subgroups shared by different 0 10 0 explicit form is givenH by 2CðIIÞ KD ¼ II\H representations. We have shown that, if we fix any repre-

Acta Cryst. (2014). A70, 417–428 Emilio Zappa et al.  Subgroup structure of the hyperoctahedral group 425 research papers

sentation in the class and a maximal subgroup P of , then Cyclic group C5 [ exp 2i=5 ]: there existsH exactly one other representation in theH class ¼ ð Þ such that P . As explained in the IntroductionK , this can be used¼H\K to describe transitions which keep intermediate symmetry encoded by P. In particular, this result implies in this context that a transition from a structure arising from via projection will result in a structure obtainable for H via projection if the transition has intermediate symmetryK described by P. Therefore, this setting is the starting point to Dihedral group D4 (the Klein Four Group): analyse structural transitions between icosahedral quasicrys- tals, following the methods proposed in Kramer (1987), Katz (1989) and Indelicato et al. (2012), which we are planning to address in a forthcoming publication. These mathematical tools also have many applications in other areas. A prominent example is virology. Viruses package Cyclic group C [! exp 2i=3 ]: their genomic material into protein containers with regular 3 ¼ ð Þ structures that can be modelled via lattices and group theory. Structural transitions of these containers, which involve rear- rangements of the protein lattices, are important in rendering certain classes of viruses infective. As shown in Indelicato et al. (2011), such structural transitions can be modelled using projections of six-dimensional icosahedral lattices and their Cyclic group C2: symmetry properties. The results derived here therefore have a direct application to this scenario, and the information on the subgroup structure of the class of crystallographic repre- sentations of the icosahedral group and their intersections provides information on the symmetries of the capsid during the transition.

APPENDIX B Here we show the explicit forms of , the representations KG in B6 of the subgroups of , together with their decomposi- APPENDIX A tions in GL 6; R . I In order to render this paper self-contained, we provide the ð Þ character tables of the subgroups of the icosahedral group, 000 001 010000 following Artin (1991), Fulton & Harris (1991) and Jones 000 010 000100 (1990). 0 00 10001 0 00000À 1 1 À ; À ; Tetrahedral group [! exp 2i=3 ]: KT ¼ B 000 100C B 1000 0 0C T ¼ ð Þ *B À C BÀ C+ B 010 000C B 001000C B C B C B 100 000C B 000010C B C B À C @ A @ A

00000 1 000001 0 1000À 0 010000 Dihedral group 10: 0 00010À 01 0 0000 101 D ; À ; KD10 ¼ B 00100 0C B 100000C *B C BÀ C+ B 000010C B 000100C B À C B C B 10000 0C B 001000C BÀ C B C @ A @ A

000001 000001 Dihedral group 6 (isomorphic to the symmetric group S3): D 0 1000À 0 000100 0 000100À 1 0 0 100001 ; À ; KD6 ¼ B 001000C B 00 1000C *B C B À C+ B 000010C B 100000C B À C B C B 100000C B 000010C BÀ C B C @ A @ A

426 Emilio Zappa et al.  Subgroup structure of the hyperoctahedral group Acta Cryst. (2014). A70, 417–428 research papers

000001 010000 0 0000 101 À ; KC5 ¼ B 100000C *BÀ C+ B 000100C B C B 001000C B C @ A 000001 000 001 0 1000À 0 000 010 0 000100À 1 0 00 10001 ; À : KD4 ¼ B 001000C B 000 100C *B C B À C+ B 000010C B 010 000C B À C B C B 100000C B 100 000C BÀ C B C @ A @ A 0 0 0001 0 0 0100 0 0 100001 À ; KC3 ¼ B 00 1000C *B À C+ B 1 0 0000C B C B 0 0 0010C B C @ A 000 001 000 010 0 00 10001 À : KC2 ¼ B 000 100C *B À C+ Figure 2 B 010 000C Algorithm 1. B C B 100 000C B C @ A 2T; 2A2 E1 E2; 2A2 2E; 2A E1 E2; KT ’ KD10 ’ È È KD6 ’ È KC5 ’ È È

2B1 2B2 2B3; 2A 2E; 2A 4B: KD4 ’ È È KC3 ’ È KC2 ’ È

APPENDIX C In this Appendix we show our algorithms, which have been implemented in GAP and used in various sections of the paper. We list them with a number from 1 to 5. Algorithm 1 (Fig. 2): Classification of the crystallographic representations of (see 4). The algorithm carries out steps Figure 3 1–4 used to proveI Propositionx 4.1. In the GAP computation, Algorithm 2. the class ^ is indicated as CB6s60. Its size is 192. B6 AlgorithmC ðIIÞ 2 (Fig. 3): Computation of the vertex star of a given vertex i in the -graphs. In the following, H stands for the class ^ of theG crystallographic representations of , B6 i 1; ...C; 192ðIIÞdenotes a vertex in the -graph correspondingI to2f the representationg H i and n stands forG the size of : we can use the size instead of the½ Š explicit form of the subgroupG since, in the case of the icosahedral group, all the non isomorphic subgroups have different sizes. Algorithm 3 (Fig. 4): Computation of the adjacency matrix of the -graph. AlgorithmG 4 (Fig. 5): This algorithm carries out a breadth- first search strategy for the computation of the connected component of a given vertex i of the -graph. G Algorithm 5 (Fig. 6): Computation of all connected Figure 4 components of a -graph. Algorithm 3. G

Acta Cryst. (2014). A70, 417–428 Emilio Zappa et al.  Subgroup structure of the hyperoctahedral group 427 research papers

Figure 6 Algorithm 5.

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428 Emilio Zappa et al.  Subgroup structure of the hyperoctahedral group Acta Cryst. (2014). A70, 417–428