Mc Laughlin Group (mathematics)

Mc Laughlin 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.