# ATLAS: Conway group Co1

Order = 4157776806543360000.
Mult = 2.
Out = 1.

### Standard generators

Standard generators of the Conway group Co1 are a and b where a is in class 2B, b is in class 3C, ab has order 40. and ababb has order 6.
Standard generators of the double cover 2Co1 are preimages A and B where B has order 3, and ABABABABBABABBABB has order 33.

### Black box algorithms

To find standard generators for Co1:
• Find any element of order 26 or 42. It powers up to a 2B-element x.
• Find any element of order divisible by 9 (i.e. 9, 18 or 36). It powers up to a 3C-element y.
• Find a conjugate a of x and a conjugate b of y, whose product has order 40, such that ababb has order 6.

### Representations

The representations of Co1 available are
• a and b as 24 x 24 matrices over GF(2).
• a and b as 276 x 276 matrices over GF(3).
• a and b as 276 x 276 matrices over GF(5).
• a and b as 276 x 276 matrices over GF(7).
• a and b as 276 x 276 matrices over GF(11).
• a and b as 276 x 276 matrices over GF(13).
• a and b as 276 x 276 matrices over GF(23).
The representations of 2Co1 available are
• A and B as 24 x 24 matrices over GF(3).
• A and B as 24 x 24 matrices over GF(5).
• A and B as 24 x 24 matrices over GF(7).
• A and B as 24 x 24 matrices over GF(11).
• A and B as 24 x 24 matrices over GF(13).
• A and B as 24 x 24 matrices over GF(23).
• A and B as 24 x 24 matrices over Z. (Also available in GAP format: A and B .)

### Maximal subgroups

The maximal subgroups of Co1 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.