# ATLAS: Fischer group F24

Order = 1255205709190661721292800.
Mult = 3.
Out = 2.

### Standard generators

Standard generators of the Fischer group F24' are a and b where a is in class 2A, b is in class 3E, ab has order 29, and abababb has order 33.
Standard generators of the triple cover 3F24 are pre-images A and B where A has order 2, and AB has order 29.
Standard generators of the automorphism group F24':2 are c and d where c is in class 2C, d is in class 8D, cd has order 29.
Standard generators of 3F24:2 are preimages C and D, where D ???.
A pair of generators conjugate to a, b can be obtained as
a' = (cdd)^{10}, b' = (cdcdd)^{-10}(cdcdcddcdcdd)^8(cdcdd)^{10}.

### Representations

The representations of F24' available are
• a and b as 781 x 781 matrices over GF(3).
The representations of 3F24' available are
• A and B as 783 x 783 matrices over GF(4).
• A and B as permutations on 920808 points.
The representations of F24':2 available are
• c and d as 781 x 781 matrices over GF(3).
• c and d as permutations on 306936 points.
The representations of 3F24:2 available are
• C and D as 1566 x 1566 matrices over GF(2).

### Maximal subgroups

The maximal subgroups of the simple group F24' are
The maximal subgroups of the automorphism group F24':2 are

### Conjugacy classes

A set of generators for the maximal cyclic subgroups up to automorphism can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements, or their images under an outer automorphism. Problems of algebraic conjugacy are also not dealt with.