McLaughlin group (mathematics)

For the television program, see The McLaughlin Group.

In the mathematical discipline known as group theory, the McLaughlin group McL is a sporadic simple group of order 2⁷ · 3⁶ · 5³· 7 · 11 = 898,128,000, discovered by 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 so is a subgroup of the Conway groups. 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 an interesting 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.

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

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

There are 12 conjugacy classes of maximal subgroups.^[1]

• U₄(3) order 3,265,920 index 275
• M₂₂ order 443,520 index 2,025
• M₂₂
• 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
• 2⁴:A₇
• M₁₁ order 7,920 index 113,400
• 5₊¹⁺²:3:8 order 3,000 index 299,376

The 2 pairs of isomorphic classes both fuse in the larger group McL:2.

References

1. ^ Atlas of finite group representations.

• 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.
• Atlas of Finite Group Representations: contains representations and other data for many finite simple groups, including the sporadic groups. Robert A. Wilson et al.
• McLaughlin, Jack (1969), "A simple group of order 898,128,000", in Brauer, R.; Sah, Chih-han, Theory of Finite Groups (Symposium, Harvard Univ., Cambridge, Mass., 1968), Benjamin, New York, pp. 109–111, MR0242941 
• Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62778-4, MR1707296 
• Wilson, Robert A. (2009), The finite simple groups, Graduate Texts in Mathematics 251, 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 05622792 

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