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McLaughlin sporadic group

Sporadic simple group

In the area of modern algebra known as group theory, the McLaughlin group McL is a sporadic simple group of order : 898,128,000 = 27 ⋅ 36 ⋅ 53 ⋅ 7 ⋅ 11 : ≈ 9.

History and properties

McL is one of the 26 sporadic groups and was discovered by as an index 2 subgroup of a rank 3 permutation group acting on the McLaughlin graph with vertices. It fixes a 2-2-3 triangle in the Leech lattice and thus is a subgroup of the Conway groups \mathrm{Co}_0, \mathrm{Co}_2, and \mathrm{Co}_3. Its Schur multiplier has order 3, and its outer automorphism group has order 2. The group 3.McL:2 is a maximal subgroup of the Lyons group.

McL has one conjugacy class of involution (element of order 2), whose centralizer is a maximal subgroup of type 2.A8. This has a center of order 2; the quotient modulo the center is isomorphic to the alternating group A8.

Representations

In the Conway group Co3, McL has the normalizer McL:2, which is maximal in Co3.

McL has 2 classes of maximal subgroups isomorphic to the Mathieu group M22. An outer automorphism interchanges the two classes of M22 groups. This outer automorphism is realized on McL embedded as a subgroup of Co3.

A convenient representation of M22 is in permutation matrices on the last 22 coordinates; it fixes a 2-2-3 triangle with vertices the origin and the type 2 points and '. The triangle's edge is type 3; it is fixed by a Co3. This M22 is the monomial, and a maximal, subgroup of a representation of McL.

(p. 207) shows that the subgroup McL is well-defined. In the Leech lattice, suppose a type 3 point v is fixed by an instance of \mathrm{Co}_3. Count the type 2 points w such that the inner product v·w = 3 (and thus v-w is type 2). He shows their number is and that this Co3 is transitive on these w.

|McL| = |Co3|/552 = 898,128,000.

McL is the only sporadic group to admit irreducible representations of quaternionic type. It has 2 such representations, one of dimension 3520 and one of dimension 4752.

Maximal subgroups

found the 12 conjugacy classes of maximal subgroups of McL as follows:

No.	Structure	Order
1	U4(3)	3,265,920
= 27·36·5·7	275
= 52·11	point stabilizer of its action on the McLaughlin graph
2,3	M22	443,520
= 27·32·5·7·11	2,025
= 34·52	two classes, fused by an outer automorphism
4	U3(5)	126,000
= 24·32·53·7	7,128
= 23·34·11
5	31+4:2.S5	58,320
= 24·36·5	15,400
= 23·52·7·11	normalizer of a subgroup of order 3 (class 3A)
6	34:M10	58,320
= 24·36·5	15,400
= 23·52·7·11
7	L3(4):22	40,320
= 27·32·5·7	22,275
= 34·52·11
8	2.A8	40,320
= 27·32·5·7	22,275
= 34·52·11	centralizer of involution
9,10	24:A7	40,320
= 27·32·5·7	22,275
= 34·52·11	two classes, fused by an outer automorphism
11	M11	7,920
= 24·32·5·11	113,400
= 23·34·52·7	the subgroup fixed by an outer involution
12	5:3:8	3,000
= 23·3·53	299,376
= 24·35·7·11	normalizer of a subgroup of order 5 (class 5A)

Conjugacy classes

Traces of matrices in a standard 24-dimensional representation of McL are shown. The names of conjugacy classes are taken from the Atlas of Finite Group Representations.

Cycle structures in the rank 3 permutation representation, degree 275, of McL are shown.

Class	Centralizer order	No. elements	Trace	Cycle type
1A	898,128,000	1	24
2A	40,320	22275 = 34 ⋅ 52 ⋅ 11	8	135, 2120
3A	29,160	30800 = 24 ⋅ 52 ⋅ 7 ⋅ 11	-3	15, 390
3B	972	924000 = 25 ⋅ 3 ⋅ 53 ⋅ 7 ⋅ 11	6	114, 387
4A	96	9355500 = 22 ⋅ 35 ⋅ 53 ⋅ 7 ⋅ 11	4	17, 214, 460
5A	750	1197504 = 26 ⋅ 35 ⋅ 7 ⋅ 11	-1	555
5B	25	35925120 = 27 ⋅ 36 ⋅ 5 ⋅ 7 ⋅ 11	4	15, 554
6A	360	2494800 = 24 ⋅ 34 ⋅ 52 ⋅ 7 ⋅ 11	5	15, 310, 640
6B	36	24948000 = 25 ⋅ 34 ⋅ 53 ⋅ 7 ⋅ 11	2	12, 26, 311, 638
7A	14	64152000 = 26 ⋅ 36 ⋅ 53 ⋅ 11	3	12, 739
7B	14	64152000 = 26 ⋅ 36 ⋅ 53 ⋅ 11	3	12, 739
8A	8	112266000 = 24 ⋅ 36 ⋅ 53 ⋅ 7 ⋅ 11	2	1, 23, 47, 830
9A	27	33264000 = 27 ⋅ 33 ⋅ 53 ⋅ 7 ⋅ 11	3	12, 3, 930
9B	27	33264000 = 27 ⋅ 33 ⋅ 53 ⋅ 7 ⋅ 11	3	12, 3, 930
10A	30	29937600 = 26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 11	3	57, 1024
11A	11	81648000 = 27 ⋅ 36 ⋅ 53 ⋅ 7	2	1125
11B	11	81648000 = 27 ⋅ 36 ⋅ 53 ⋅ 7	2	1125
12A	12	74844000 = 25 ⋅ 35 ⋅ 53 ⋅ 7 ⋅ 11	1	1, 22, 32, 64, 1220
14A	14	64152000 = 26 ⋅ 36 ⋅ 53 ⋅ 11	1	2, 75, 1417
14B	14	64152000 = 26 ⋅ 36 ⋅ 53 ⋅ 11	1	2, 75, 1417
15A	30	29937600 = 26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 11	2	5, 1518
15B	30	29937600 = 26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 11	2	5, 1518
30A	30	29937600 = 26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 11	0	5, 152, 308
30B	30	29937600 = 26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 11	0	5, 152, 308

Generalized Monstrous Moonshine

Main article: Generalized moonshine

Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For the Conway groups, the relevant McKay–Thompson series is T_{2A}(\tau) and T_{4A}(\tau).

References

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: "Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups." Oxford, England 1985.

References

Conway et al. (1985)
"ATLAS: MCL — Permutation representation on 275 points".
"ATLAS: MCL — Permutation representation on 275 points".

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