# McLaughlin group (mathematics)

In the mathematical 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 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 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.

## 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 $T_{2A}(\tau)$ and $T_{4A}(\tau)$.

Let $w_n$ = 1, 23, 253, 253, 275, 896, 896, 1771, 2024,...() and $x_n$ = 1, 22, 231, 252, 770, 770, 896, 896,...() be the degrees of irreducible representations of Co3 and McL, respectively. Then,

\begin{align} 1 &= w_1 = x_1\\ 23 &= w_2 = x_1+x_2\\ 253 &= w_3 = x_2+x_3\\ 253 &= w_4 = x_2+x_3\\ 275 &= w_5 = x_1+x_2+x_4 \\ 896 &= w_6 = x_7 \\ \end{align}

which implies there is moonshine between McL and the relevant McKay-Thompson series as well.

## Maximal subgroups

Finkelstein (1973) showed that there are 12 conjugacy classes of maximal subgroups as follows:

• U4(3) order 3,265,920 index 275
• 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