Talk:McLaughlin sporadic group

Representation

The Mathieu group M₂₂ occurs as a maximal subgroup of McL in 2 ways. One permutation matrix representation of M₂₂ fixes a 2-2-3 triangle with vertices (-3,1²³), (4,4,0²²), and -(1,5,1²²). A non-monomial matrix is needed to complete generation of a representation of McL. Scott Tillinghast, Houston TX (talk) 16:04, 21 December 2015 (UTC)

2-2-3 triangle has vertices (-3,1²³), -(4,4,0²²), and the origin, with a side (1,5,1²²).

This representation has a conveniently monomial maximal subgroup M₂₂. The full McL cannot be a subgroup of the monomial group 2¹²:M₂₄ because the latter has no element of order 9. Group theorists seem to be loathe to publish about needed non-monomial matrices. Scott Tillinghast, Houston TX (talk) 14:05, 17 April 2018 (UTC)

Centralizer of involution

M₂₂ and McL both have just one conjugacy class of involution. In both groups a Sylow 2-subgroup has order 128.

In M₂₂ the centralizer of an involution has order 384=3*128. In McL this expands to the double cover 2.A₈, which has order 3*5*7*384.

Co₀ has another conjugacy class of subgroup isomorphic to the double cover 2.A₈, commuting with a permutation of shape 3⁸, with a different center. Co₀ also has subgroups isomorphic to A₈. Scott Tillinghast, Houston TX (talk) —Preceding undated comment added 19:32, 26 December 2015 (UTC)