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 Co0, Co2 and Co3. 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 . 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:
+ Maximal subgroups of McL No. Structure Order Index Comments 1 U4(3) 3,265,920= 27·36·5·7 275= 52·11 point stabilizer of its action on the McLaughlin graph2,3 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 involution9,10 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= 23·34·52·7the 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. Conway et al. (1985) 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.
ClassCentralizer orderNo. elementsTraceCycle 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 power equivalent 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 power equivalent 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 power equivalent 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 power equivalent 14B 14 64152000 = 26 ⋅ 36 ⋅ 53 ⋅ 11 1 2, 75, 1417 15A 30 29937600 = 26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 11 2 5, 1518 power equivalent 15B 30 29937600 = 26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 11 2 5, 1518 30A 30 29937600 = 26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 11 0 5, 152, 308 power equivalent 30B 30 29937600 = 26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 11 0 5, 152, 308
Generalized Monstrous 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 and .
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.
External links MathWorld: McLaughlin group Atlas of Finite Group Representations: McLaughlin group