ATLAS: McLaughlin group McL

Order = 898128000 = 27.36.53.7.11.
Mult = 3.
Out = 2.

Standard generators

Standard generators of the McLaughlin group McL are a and b where a is in class 2A, b is in class 5A, ab has order 11, and ababababbababbabb has order 7.
Standard generators of the triple cover 3.McL are pre-images A and B where A has order 2, and B has order 5.

The outer automorphism is achieved by this program.

Standard generators of the automorphism group McL:2 are c and d where c is in class 2B, d is in class 3B, cd has order 22, and cdcdcdcddcdcddcdd has order 24.
Standard generators of 3.McL:2 are pre-images C and D where CDCDCDDCD has order 11.
A pair of generators conjugate to a, b can be obtained as
a' = (cd)^{-1}(cdcdcddcdcdcddcd)^{12}cd, b' = (cdd)^{-3}(cdcdd)^{3}(cdd)^3.


Black box algorithms

To find standard generators for McL: To find standard generators for McL.2:

Representations

The representations of McL available are The representations of 3.McL available are The representations of McL:2 available are The representations of 3.McL:2 available are

Maximal subgroups

The maximal subgroups of McL are The maximal subgroups of McL:2 are

Conjugacy classes

A set of generators for the maximal cyclic subgroups can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. Problems of algebraic conjugacy are not yet dealt with.
Main ATLAS page Return to main ATLAS page.

Last updated 10th December 1998,
R.A.Wilson, R.A.Parker and J.N.Bray