# ATLAS: Conway group Co3

Order = 495766656000 = 210.37.53.7.11.23.
Mult = 1.
Out = 1.

### Standard generators

Standard generators of the Conway group Co3 are a and b where a is in class 3A, b is in class 4A, and ab has order 14.

### Black box algorithms

To find standard generators for Co3:
• Find any element of order 9, 18, 24 or 30. It powers up to a 3A-element x.
• Find any element of order 20. It powers up to a 4A-element y.
• Find a conjugate a of x and a conjugate b of y such that ab has order 14.

### Representations

The representations of Co3 available are:
• a and b as permutations on 276 points.
• a and b as permutations on 552 points - imprimitive.
• a and b as permutations on 11178 points.
• a and b as permutations on 37950 points.
• a and b as permutations on 48600 points.
• a and b as permutations on 128800 points. (Reconstructed on 14/9/99.)
• a and b as 22 × 22 matrices over GF(2).
• a and b as 22 × 22 matrices over GF(3).
• a and b as 126 × 126 matrices over GF(3).
• a and b as 126 × 126 matrices over GF(3) - the dual of the above.
[NB: The ordering of the representations of degree 126 over GF(3) has not been finalised.]
• a and b as 23 × 23 matrices over GF(5).
• a and b as 23 × 23 matrices over GF(7).
• a and b as 23 × 23 matrices over GF(11).
• a and b as 23 × 23 matrices over GF(23).
• a and b as 23 × 23 matrices over Z.

### Maximal subgroups

The maximal subgroups of Co3 are as follows. Words provided by Peter Walsh, implemented and checked by Ibrahim Suleiman.

### 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.