It fixes a 2-2-3 triangle in the Leech lattice and thus is a subgroup of the Conway groups
McL has one conjugacy class of involution (element of order 2), whose centralizer is a maximal subgroup of type 2.A8.
McL has 2 classes of maximal subgroups isomorphic to the Mathieu group M22.
A convenient representation of M22 is in permutation matrices on the last 22 coordinates; it fixes a 2-2-3 triangle with vertices the origin and the type 2 points x = (−3, 123) and y = (−4,-4,022)'.
McL is the only sporadic group to admit irreducible representations of quaternionic type.
[1] The names of conjugacy classes are taken from the Atlas of Finite Group Representations.
[2] Cycle structures in the rank 3 permutation representation, degree 275, of McL are shown.
[3] 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.