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)