Math. J. Okayama Univ. 55 (2013), 145–155

MULTIPLICITY-FREE PERMUTATION CHARACTERS OF
COVERING GROUPS OF SPORADIC SIMPLE GROUPS
S. A. LINTON and Z. E. MPONO

Abstract. In this paper we classify all multiplicity-free faithful permutation representations of the covering groups of the sporadic simple
groups. These results were obtained computationally, making extensive
use of the GAP library of character tables.

1. Introduction
Any permutation representation of a group G can be extended to a linear
representation (on a space with basis in bijection with the set permuted)| . One can thenover any field, in particular over the complex numbers C
consider the decomposition of this linear representation into its irreducible
constituents. The structure of this decomposition is linked to the structureof the permutation representation. For instance the multiplicity of the trivial
character gives the number of orbits, and, for transitive representations, thesum of the squares of the multiplicities of the constituents gives the rank(the number of orbits of the point stabilizer).
Of particular interest are those cases where the multiplicities of all theconstituents are one, called multiplicity-free representations. This condition
is related to two natural permutation-theoretic conditions as follows:
generously transitive=⇒

multiplicity-free=⇒transitive(cf. [2] p63). Multiplicity-free permutation representations can also be characterized as those whose centralizer algebra is abelian [5]. Also, the actionof the automorphism group on the vertices of any distance transitive graphis always multiplicity-free, so that a classification of multiplicity-free permutation representations serves as an initial step towards a classification of
distance-transitive graphs with a given automorphism group. General results on the classification of distance-transitive graphs [6] enable many cases
to be reduced to graphs whose automorphism groups are simple or almost
simple (simple modulo a cyclic central subgroup and/or some extending
automorphisms).
Mathematics Subject Classification. Primary 20C; Secondary: 20C34, 20D08.
Key words and phrases. multiplicity-free faithful permutation representations, coveringgroups of the sporadic simple groups.
The second author was supported by a REDIBA grant from the University of South
Africa (UNISA) and the National Research Foundation (NRF) of South Africa.
145

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S. A. LINTON AND Z. E. MPONO

The classification of Finite Simple Groups arranges all simple groups into
four families: cyclic groups of prime order; alternating groups; finite groupsof Lie type and the 26 sporadic groups, and it is this last family which
is our focus here. The multiplicity-free permutation representations of thesporadic groups and their automorphism groups are completely known [5, 1]
and tabulated in [1].
This paper is intended as a sequel to [1]. We give the multiplicity-free,faithful permutation characters of the covering groups m·G of the sporadic
simple groups, where G is a sporadic simple group and m divides the orderof its Schur multiplier (given, for instance, in the Atlas of Finite Groups [3]).
The following sporadic groups have trivial Schur multipliers and so do notpossess nontrivial covering groups: M11 , J1 , M23 , M24 , He, Co3 , Co2 , HN ,Ly, T h, F i23 , J4 and M .
In Section 2, we give some background and prove some results whichhave been used to obtain the desired permutation characters. We also deal
with the computational techniques that have been used in determining themultiplicity-free and faithful representations. In Section 3, we present theresults in tables. In Section 4, we give a more detailed explanation of certain
complex cases and prove the one result for which our general methods were
inadequate.
2. Preliminaries
Proposition 2.1. Let G be a group of shape m·G and φ : G −→ G thenatural homomorphism from G to G ∼= G/Z, where Z is the centre of G.
If G has a multiplicity-free faithful permutation representation with point
stabilizer H, where H = Hφ then the permutation action of G on the cosetsof H is also multiplicity-free and faithful.
Proof. The action of G on the cosets of H is equivalent to that of G on
the cosets of HZ (on which Z will act trivially). The bijection is provided
by ψ : HZx −→ Hxφ, which is easily checked to be well-defined, surjective
and injective. This, with φ is an equivalence of permutation representations,since
HZxyψ = H(xy)φ = Hxφyφ = HZxψyφ.
Now in the space V with basis defined on the cosets of H in G, consider thesubspace W spanned by the vectors
X
wx =
Hxz
z∈Z

where x ∈ G. Two such vectors wx and wy are equal if and only if x and ylie in the same coset of HZ, and since Z is central in G it is easy to checkMULTIPLICITY-FREE PERMUTATION CHARACTERS

147

that W is invariant under G, furthermore, the action of G on W is just thaton the space spanned by the cosets of HZ.
Thus the permutation representation of G on H appears as a submoduleof that of G on H and so, if the latter is multiplicity-free and faithful, thenthe former certainly must be.
Proposition 2.1 above allows us to take as the starting point for ourclassification, the classification in [1] of multiplicity-free permutation representations for the sporadic groups, and consider in each case its possible
extensions.
We now consider a candidate permutation representation of a simple
group G with point stabilizer H and consider how it might extend to afaithful representation of G. This is determined by the subgroup K = Hφ−1of G. If the permutation representation extends, then the point stabilizer
H ≤ G will certainly be a subgroup of K. Furthermore, we must have
Hφ = H and H ∩ Z = {1}. In fact there will be one equivalence class ofextension for each G−conjugacy class of such subgroup in K. Note that, in
this situation φ actually provides an isomorphism between H and H.
Remark 2.2. If Z has a proper subgroup Y , then G/Y will also be a covering group of G and it is easy to check that a (multiplicity-free) permutation
representation of G can only extend to a faithful (multiplicity-free) permutation representation of G if it also extends to a faithful (multiplicity-free)permutation representation of G/Y .
In all the cases we will consider, the central subgroup Z of G is cyclicof order m, say. Let z be a generator of Z. We can obtain useful further
information about the decomposition of a permutation representation byconsidering the action of z. Let V be the complex vector space on which
G acts by permuting the basis. Then V must decompose into eigenspaces:
E0 , E1 , . . . , Em−1 where z acts as η i on Ei (η a primitive mth root of unity).
We can make various observations about this decomposition.
• In a transitive faithful permutation action, no nonidentity elementof Z can fix any points sinceαz = α =⇒ αgz = αzg = αg
which, with transitivity, would imply that z fixes every point, contradicting faithfulness. Thus z acts as a product of m-cycles.
• Let (α0 , α1 , . . . , αm−1 ) be a cycle of z. Then it is easy to see that
m−1
X
i=0

αi η ij ∈ Ej .

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S. A. LINTON AND Z. E. MPONO

From this, we can see that dimEi = dimV /m.
• Since Z is central, each Ei is invariant under G.
• The action of G on Ei has kernel generated by z (i,m) and can containno irreducible constituent which is not faithful modulo this kernel.
• Finally, complex conjugation obviously exchanges Ei and E−i (indices taken modulo m). Since the action of G on V is given byreal (permutation) matrices, however, complex conjugation must
preserve the G-module structure of the Ei , so whatever irreducible
constituents make up Ei , their complex conjugates must make up
E−i .
• The subspace W considered above is simply E0 .
In the particular cases we are considering, these observations have thefollowing consequences:• when m = 2, V is the direct sum of two components of equal dimension, one composed entirely of characters of G, the other of faithfulcharacters of 2·G• when m = 3, V is the direct sum of three components of equal dimension, one composed entirely of characters of G, the other twocomplex conjugates of one another and composed entirely of faithfulcharacters of 3·G• when m = 4, V is the direct sum of four components of equal dimension, one composed entirely of characters of G, one composed offaithful characters of 2·G and the other two complex conjugates of
one another and composed entirely of faithful characters of 4·G• when m = 6, V is the direct sum of six components of equal dimension, one composed entirely of characters of G, one composedof faithful characters of 2·G and the others making up two pairs ofcomplex conjugates, one composed of faithful characters of 3·G and
the other of 6·G• when m = 12, V is the direct sum of twelve components of equal
dimension, one composed entirely of characters of G, one composedof faithful characters of 2·G and the others making up five pairs of

MULTIPLICITY-FREE PERMUTATION CHARACTERS

149

complex conjugates, one composed of faithful characters of 3·G, oneof 4·G, one of 6·G and the other two of 12·G
Remark 2.2 above implies that when m is composite, we need only consider extensions of representations that have already extended to k·G whenk divides m.
Our main techniques for obtaining these results are character-theoretic,
and make extensive use of the character table library of the GAP [4] system,
which in particular includes the character tables of all the sporadic simple
groups, their covering groups and many of their maximal subgroups.
In the smaller cases, we make use of the PermChars function of GAP
to list all combinations of irreducible characters which have the correct degree, and pass a number of tests for being permutation characters. In [1],all characters satisfying the properties of [1, Lemma 4.3] are called possible
permutation characters. We have to be aware of two pitfalls here: firstly, thetests are not complete and some characters pass them all, but are not characters of any permutation representations; secondly, inequivalent permutation
representations may give rise to the same character. Nevertheless, if there
are no multiplicity-free candidate permutation characters, then we knowthat there is no multiplicity-free permutation representation and if there isjust one candidate permutation character, and we can exhibit a subgroupof that index, then we know that the corresponding action must afford thatcharacter. Furthermore faithfulness of a representation of a covering groupcan be checked both in the character (which must vanish on nonidentity
elements of the centre) and from the subgroup, which must meet the centre
only trivially. Thus, if we exhibit a faithful representation, and have justone faithful character, we again know that they must correspond.
Multiplicity-free permutation characters can be determined using the tables of marks. [1, Appendix 1] gives a GAP procedure for computing
multiplicity-free permutation characters using the tables of marks.
In the larger cases, we use character induction, since in all but one case,the character table of the appropriate subgroups are available (see Section
4 for the details of this one case). Here we use the PossibleClassFusions
function of GAP to find all possible embeddings of H into G and thencompute, for each one, the character 1H ↑G . One of these would have to be
the character of any permutation representation with point stabilizer isomorphic to H. In practice, sufficiently few possible class fusions were foundin each case, that there was no problem identifying the correct permutation
character(s).
[1, Lemma 4.4] describes a technique for determining all multiplicityfree characters χ ↑G for χ a possible permutation character of a maximal

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S. A. LINTON AND Z. E. MPONO

subgroup H of a group G. The corresponding GAP procedure for thistechnique is given in [1, Appendix 2].
3. Results
We present the results in tables with various columns. The first column
describes the isomorphism type of the point stabilizer subgroup, which, as
noted above, must be the isomorphism type of the corresponding point stabilizer in the simple group.
The second column refers to the index in m·G of the group in column
1. In tables for 2·G and 3·G, the third column refers to the line numberin the relevant table in [1] which describes the corresponding permutation
representation of the simple group G. In tables for 6·G, the third and fourthcolumns give the row numbers of the corresponding representations in thetables for 2·G and 3·G respectively. The next column gives the decomposition of the faithful part of the permutation module. The decompositions of
the non-faithful parts are found in the indicated rows of the tables for G (in[1]), 2·G and 3·G. In giving these decompositions, we denote by e.g 123athe first faithful representation of degree 123 in the Atlas [3]. In the case of
3·G or 6·G where each row of the Atlas table corresponds to two complex
conjugate characters (indicated there by ◦2), we call the first e.g 123b andthe other 123b′ . The last column gives the rank of the permutation action.
Other notations follow the same format and style as used in [1].
All computations were performed using GAP 4 running on a Pentium
III computer at the Centre for Interdisciplinary Research in Computational
Algebra at the University of St Andrews.
The following covers of sporadic simple groups have no multiplicity-freepermutation representations and therefore no corresponding tables below:
3·J3 , 4·M22 , 12·M22
2·M12
M11
24 1 +12a3
M11
24 2 +12a3
A6 ·21 264 5 +12a + 120a
7
A6 ·21 264 8 +12a + 120a7
32 ·2·S4 440 11 +110ab
72
3 ·2·S4 440 13 +110ab
72
3 ·2·S4 440 11 +12a + 44ab + 120a92
3 ·2·S4 440 13 +12a + 44ab + 120a932 ·2·A4 880 12 +12a + 44ab + 110ab + 120a 14
32 ·2·A4 880 14 +12a + 44ab + 110ab + 120a 14

MULTIPLICITY-FREE PERMUTATION CHARACTERS

151

2·M22
24 :A5A7
A7
3
2 :L3 (2)

924
352
352
660

3
4
5
7

+126ab + 210a
+56a + 120a
+56a + 120a
+120a + 210a

8
5
5
7

3·M22
24 :A51386 3 +21aa′ + 105aa′ bb′ + 231aa′
24 :S5
693 6 +21aa′ + 105aa′ bb′
3
2 :L3 (2) 990 7 +21aa′ + 99aa′ + 105aa′ bb′
L2 (11) 2016 9 +21aa′ + 105aa′ bb′ + 210aa′ + 231aa′

13
10
13
16

6·M22
24 :A52772 1 1 +126aa′ bb′ + 210aa′ 22
23 :L3 (2) 1980 4 3 +330aa′
17
2·J2
U3 (3) 200 1 +50ab 5
2·HS
U3 (5) 704
3
U3 (5) 704
5A84400 8M11 11200 10
M11 11200 11+176ab6+176ab6+176ab + 924ab
13+56a + 176ab + 616ab + 1980ab 16+56a + 176ab + 616ab + 1980ab 16
3·M cL

2·A8 66825 6 +2772aa′ + 5103aa′ + 6336aa′ + 8064aa′ 14
2·Ru
2 F (2)′416240 2 +28ab + 4032ab 9
2·Suz

U5 (2) 65520 4 +364ab + 16016ab 10

152

S. A. LINTON AND Z. E. MPONO

G2 (4)
U5 (2)
1+6
2− ·U4 (2)

5346
98280
40540524+6 :3A6

1216215 6

3·Suz+66aa′ + 1716aa′
7
′
′
′
′
+78aa + 1365aa + 4290aa + 27027aa
14
′
′
′
′
+66aa + 429aa + 1716aa + 6720aa
23+18954aa′ + 42900aa′ + 64350aa′+1365aa′ + 4290aa′ + 27027aa′ + 42900aa′ 27+85800aa′ + 104247aa′ + 139776aa′

1
4
5

6·Suz
U5 (2) 196560 1 2 +12aa′ + 924aa′ + 4368aa′ + 27456aa′ 26

L3 (7):2
L3 (7):2
L3 (7)
L3 (7)

3682803682807365607365601
3
2
4

3·O ′ N+495aa′ + 58653aa′ + 63612aa′+495bb′ + 58653aa′ + 63612aa′+495aa′ + 58653aa′ + 63612aa′ + 122760aa′+495bb′ + 58653aa′ + 63612aa′ + 122760aa′

11
11
15
15

2·F i22
O7 (3)
O7 (3)
O8+ (2):S3
O8+ (2):S3
O8+ (2)·3
O8+ (2)·2
O8+ (2)·2

28160
28160
123552123552247104370656370656

2
3
4
4
5
6
6

+352a + 13728a
5+352a + 13728b
5+13728a + 48048a
6+13728b + 48048b
6+13728ab + 48048ab
11+352a + 13728a + 48048a + 123200a 11+352a + 13728b + 48048b + 123200a 11
3·F i22

O8+ (2):S3
O8+ (2)·3
O8+ (2)·3
O8+ (2)·2
26 :S6 (2)2 F (2)′
4

4 +351aa′ + 19305aa′ + 42120aa′
5 +27027aa′ + 96525aa′
5 +351aa′ + 7722aa′ + 19305aa′+42120aa′ + 54054aa′
555984 6 +351aa′ + 19305aa′ + 27027aa′+42120aa′ + 96525aa′
2084940 8 +351aa′ + 19305aa′ + 27027aa′ + 42120aa′+96525aa′ + 123552aa′ + 386100aa′
10777536 9 +19305aa′ + 27027aa′ + 51975aa′+386100aa′ + 405405aa′ + 1351350cc′ dd′
185328370656370656

10
11
17
17
24
25

MULTIPLICITY-FREE PERMUTATION CHARACTERS

O8+ (2):S3
O8+ (2):S3
O8+ (2)·3
O8+ (2)·2
O8+ (2)·2

Co2Co337065637065674131211119681111968345676·F i22
1 +61776aa′
1 +61776bb′
3 +61776aa′ bb′
4 +61776aa′ + 123552aa′
4 +61776bb′ + 123552aa′

153

14
14
25
25
25

2·Co1
196560 1 +24a + 2576a + 95680a
716773120 5 +24a + 2576a + 95680a + 1841840a 12+6446440a
3·F i′24F i23920808
1 +783aa′ + 306153aa′−
O10(2) 150532080426 2 +783aa′ + 64584aa′ + 306153aa′+6724809aa′ + 19034730aa′ + 43779879aa′+195019461aa′ + 203843871aa′+1050717096aa′ + 1818548820aa′+10726070355aa′ + 15016498497aa′+21096751104aa′

7
43

2·B
F i23

2031941058560000 4 +96256a + 10506240a + 410132480a
34+8844386304a + 36657653760a+864538761216a + 4322693806080a+10177847623680a + 60780833777664a+110949141022720a + 828829551513600a−
Except for the two largest ones (3·F i′24 on O10(2) and 2·B on F i23 ),explicit permutations have been constructed for all of these representations(some were already known) and are available from the first author.

4. Remarks on specific cases
There are two cases in the above tables where the results reflect especiallycomplex situations in the groups concerned, and one where a special proof
technique was needed.
According to [1], M12 has two classes of maximal subgroups isomorphic to
A6 ·22 , two classes of non-maximal subgroups isomorphic to A6 ·21 ∼= S6 and∼two classes isomorphic to A6 ·22 = P GU2 (9). In each case, the two classes
are exchanged by the outer automorphism of M12 and each class affords
a multiplicity-free permutation character. It is easy to see that M12 must154

S. A. LINTON AND Z. E. MPONO

also have two classes of subgroups isomorphic to A6 ·23 ∼= M10 , but these do
not afford a multiplicity-free permutation character. Again examining [1]we see that the two conjugacy classes of subgroups isomorphic to A6 ·21 ∼=S6 actually afford the same permutation character (although inequivalent
permutation representations).
The GAP function PermChars actually reveals five distinct multiplicityfree possible permutation characters of degree 132 for M12 , and we conclude
that two of them cannot actually be given by any permutation representations.
In 2·M12 , we find that the preimage of A6 ·22 is a group H which can be
described as (2 × A6 )·22 . The largest solvable quotient of this: H/H (∞) ,is isomorphic to D8 . Within this, we find that the preimage of a subgroup
A6 ·21 is a subgroup H1 of shape 2 × A6 ·21 while that of a subgroup of
shape A6 ·22 is H2 of shape (2 × A6 )·2 with the outer part affording the 22
automorphism and squaring to the centre.
Now H has no subgroup of index 2 not containing the centre (as may
be seen in D8 ) and so there are no faithful extensions of the permutation
action of M12 on 66 points. H1 has two conjugacy classes of subgroups ofindex 2, but they are conjugate in H, so we see just one equivalence class of
permutation representations (and its image under the outer automorphism),
which turn out to be multiplicity-free. H2 again has no subgroup of index
2 not containing the centre.
Thus we see in our table just two equivalence classes of faithful multiplicityfree permutation representations of 2·M12 with point stabilizers isomorphic
to A6 ·21 , exchanged by the outer automorphism of 2·M12 and affording thesame character.
In F i22 , there is a conjugacy class of subgroups isomorphic to O8+ (2):S3
which extends to 2 × O8+ (2):S3 in F i22 ·2. The permutation representationof F i22 on the cosets of O8+ (2):S3 is multiplicity-free.
In 2·F i22 , the preimage of a subgroup in this class is isomorphic to
2 × O8+ (2):S3 , and so has two non-conjugate subgroups of index 2 not containing the central element, leading to two multiplicity-free faithful permutation representations of 2·F i22 . However, it can be seen from the character
decompositions that these are exchanged by the outer automorphism.
Thus, the preimage of 2 × O8+ (2):S3 in (either isomorphism type of)
2·F i22 ·2 must be a group (2 × O8+ (2):S3 )·2 in which the outer involution
acting by conjugation ”shifts” elements in the outer half of 2 × O8+ (2):S3 by
the central element of that group.
Exactly the same phenomenon occurs for the non-maximal subgroup+
O8 (2)·2.

MULTIPLICITY-FREE PERMUTATION CHARACTERS155

The non-maximal subgroup O8+ (2)·3 has preimage isomorphic to 3 ×
O8+ (2)·3 in 3·F i22 . This group has two conjugacy classes of subgroups ofindex 3 not containing the centre, which give rise to inequivalent faithful
multiplicity-free permutation representations of 3·F i22 , one of rank 11 andthe other rank 17.
Finally, the character table of the maximal subgroup 21+22·Co2 of B is+
not known, so we could not use our normal character induction methods.
Instead, we look at the degrees of the faithful characters of 2·B and caneasily check that no combination of these (without repetition) can sum to11707448673375. Thus there can be no faithful multiplicity-free permutation
representation of 2·B with this stabilizer.
Acknowledgements
This work was done while the second author was on a brief visit to theCentre for Interdisciplinary Research in Computational Algebra at the University of St Andrews, per invitation by the first author who is the directorof the centre. Gratitude is hereby extended to the Livesey family for opening
the doors of their home for the second author to reside while on this brief
visit.
References[1] T. Breuer and K. Lux, The multiplicity-free permutation characters of the sporadic
simple groups and their automorphism groups, Comm. Algebra 24(7), 1996, 2293 2316.
[2] A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, SpringerVerlag, Berlin, 1989.
[3] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of
Finite Groups, Oxford University Press, Oxford, 1985.
[4] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.2; 2000,(http://www.gap-system.org)[5] A. A. Ivanov, S. A. Linton, K. Lux, J. Saxl and L. H. Soicher, Distance-transitive
representations of the sporadic groups, Comm. Algebra 23(9), 1995, 3379 - 3427.
[6] C. E. Praeger, J. Saxl and K. Yokoyama, Distance transitive graphs and finite simple
groups, Proc. London Math. Soc. 55, 1987, 1 - 21.
S. A. Linton
University of St Andrews, School of Computer Science, North Haugh
St Andrews, Fife KY16 9SS, Scotlande-mail address: sal@dcs.st-and.ac.uk
Z. E. Mpono (corresponding author)
University of South Africa, Department of Mathematical Sciences
P. O. Box 392, Unisa 0003, South Africa
e-mail address: mponoze@unisa.ac.za(Received December 15, 2010 )