McLaughlin sporadic group


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

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,, 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, 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.
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. He shows their number is and that this Co3 is transitive on these w'.
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.StructureOrderIndexComments
1U43,265,920
= 27·36·5·7		275
= 52·11		point stabilizer of its action on the McLaughlin graph
2,3M22443,520
= 27·32·5·7·11		2,025
= 34·52		two classes, fused by an outer automorphism
4U3126,000
= 24·32·53·7		7,128
= 23·34·11
531+4:2.S558,320
= 24·36·5		15,400
= 23·52·7·11		normalizer of a subgroup of order 3
634:M1058,320
= 24·36·5		15,400
= 23·52·7·11
7L3:2240,320
= 27·32·5·7		22,275
= 34·52·11
82.A840,320
= 27·32·5·7		22,275
= 34·52·11		centralizer of involution
9,1024:A740,320
= 27·32·5·7		22,275
= 34·52·11		two classes, fused by an outer automorphism
11M117,920
= 24·32·5·11		113,400
= 23·34·52·7		the subgroup fixed by an outer involution
125:3:83,000
= 23·3·53		299,376
= 24·35·7·11		normalizer of a subgroup of order 5

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.
ClassCentralizer orderNo. elementsTraceCycle type
1A 898,128,000124--
2A40,32022275 = 34 ⋅ 52 ⋅ 118135, 2120-
3A29,16030800 = 24 ⋅ 52 ⋅ 7 ⋅ 11-315, 390-
3B 972924000 = 25 ⋅ 3 ⋅ 53 ⋅ 7 ⋅ 116114, 387-
4A 969355500 = 22 ⋅ 35 ⋅ 53 ⋅ 7 ⋅ 11417, 214, 460-
5A 7501197504 = 26 ⋅ 35 ⋅ 7 ⋅ 11-1555-
5B 2535925120 = 27 ⋅ 36 ⋅ 5 ⋅ 7 ⋅ 11415, 554-
6A 3602494800 = 24 ⋅ 34 ⋅ 52 ⋅ 7 ⋅ 11515, 310, 640-
6B 3624948000 = 25 ⋅ 34 ⋅ 53 ⋅ 7 ⋅ 11212, 26, 311, 638-
7A 1464152000 = 26 ⋅ 36 ⋅ 53 ⋅ 11312, 739power equivalent
7B 1464152000 = 26 ⋅ 36 ⋅ 53 ⋅ 11312, 739power equivalent
8A 8112266000 = 24 ⋅ 36 ⋅ 53 ⋅ 7 ⋅ 1121, 23, 47, 830-
9A 2733264000 = 27 ⋅ 33 ⋅ 53 ⋅ 7 ⋅ 11312, 3, 930power equivalent
9B 2733264000 = 27 ⋅ 33 ⋅ 53 ⋅ 7 ⋅ 11312, 3, 930power equivalent
10A3029937600 = 26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 11357, 1024-
11A1181648000 = 27 ⋅ 36 ⋅ 53 ⋅ 721125power equivalent
11B1181648000 = 27 ⋅ 36 ⋅ 53 ⋅ 721125power equivalent
12A1274844000 = 25 ⋅ 35 ⋅ 53 ⋅ 7 ⋅ 1111, 22, 32, 64, 1220-
14A1464152000 = 26 ⋅ 36 ⋅ 53 ⋅ 1112, 75, 1417power equivalent
14B1464152000 = 26 ⋅ 36 ⋅ 53 ⋅ 1112, 75, 1417power equivalent
15A3029937600 = 26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 1125, 1518power equivalent
15B3029937600 = 26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 1125, 1518power equivalent
30A3029937600 = 26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 1105, 152, 308power equivalent
30B3029937600 = 26 ⋅ 35 ⋅ 52 ⋅ 7 ⋅ 1105, 152, 308power equivalent

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.