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McLaughlin sporadic group

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In the area of modern algebra known as group theory, the McLaughlin group McL is a sporadic simple group of order

   27 ⋅ 36 ⋅ 53 ⋅ 7 ⋅ 11 = 898,128,000
≈ 9×108.

History and properties

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McL is one of the 26 sporadic groups and was discovered by Jack McLaughlin (1969) as an index 2 subgroup of a rank 3 permutation group acting on the McLaughlin graph with 275 = 1 + 112 + 162 vertices. It fixes a 2-2-3 triangle in the Leech lattice and thus is a subgroup of the Conway groups , , and . 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

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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 x = (−3, 123) and y = (−4,-4,022)'. The triangle's edge x-y = (1, 5, 122) is type 3; it is fixed by a Co3. This M22 is the monomial, and a maximal, subgroup of a representation of McL.

Wilson (2009) (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 . 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 552 = 23⋅3⋅23 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

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Finkelstein (1973) found the 12 conjugacy classes of maximal subgroups of McL as follows:

  • U4(3) order 3,265,920 index 275 – point stabilizer of its action on the McLaughlin graph
  • M22 order 443,520 index 2,025 (two classes, fused under an outer automorphism)
  • U3(5) order 126,000 index 7,128
  • 31+4:2.S5 order 58,320 index 15,400
  • 34:M10 order 58,320 index 15,400
  • L3(4):22 order 40,320 index 22,275
  • 2.A8 order 40,320 index 22,275 – centralizer of involution
  • 24:A7 order 40,320 index 22,275 (two classes, fused under an outer automorphism)
  • M11 order 7,920 index 113,400
  • 5+1+2:3:8 order 3,000 index 299,376

Conjugacy classes

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Traces of matrices in a standard 24-dimensional representation of McL are shown. [1] The names of conjugacy classes are taken from the Atlas of Finite Group Representations.[2]

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

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

Generalized Monstrous Moonshine

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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 and .

References

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