McLaughlin group (mathematics)

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

McL is one of the 26 sporadic groups and was 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 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.

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. A permutation matrix representation of M₂₂ on the last 22 coordinates fixes a 2-2-3 triangle with vertices x = (1, 5, 1²²), y = (−3, 1²³), and z the origin. x is type 3 while y and Template:Norap are type 2.

In the Leech lattice, suppose a type 3 point v is fixed by an instance of Co₃. Count the type 2 points w such that inner product {{{1}}} and thus v-w is type 2. Wilson (2009) (p. 207) shows their number is 552 = 2³⋅3⋅23. He also shows this Co₃ is transitive on these w. Hence the subgroup McL is well-defined.

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