# ATLAS: Conway group Co2

Order = 42305421312000 = 218.36.53.7.11.23.
Mult = 1.
Out = 1.

The following information is available for Co2:

### Standard generators

Standard generators of the Conway group Co2 are a and b where a is in class 2A, b is in class 5A and ab has order 28.

### Black box algorithms

#### Finding generators

To find standard generators for Co2:

1. Find any element of order 16, 18 or 28. This powers up to a 2A­element x.
[The probability of success at each attempt is 155 in 1008 (about 1 in 7).]
2. Find any element of order 15 or 30, This powers up to y of order 5.
[The probability of success at each attempt is 1 in 5, and the probability that y ends up being in class 5A is 2 in 3.]
3. Find a conjugate a of x and a conjugate b of y, whose product has order 28.
[If y is in class 5A, then the probability of success at each attempt is 40 in 759 (about 1 in 19).
If y is in class 5B, then the probability of success at each attempt is 8 in 759 (about 1 in 95).]
If you have still not succeeded after (say) 35 attempts at this step, you begin to suspect that y is in the wrong conjugacy class, so go back to Step 2.
4. If abb has order 15, then y is in the wrong conjugacy class, so go back to Step 2.
5. Otherwise abb has order 9 and standard generators for Co2 have been obtained.
This algorithm is available in computer readable format: finder for Co2.

#### Checking generators

To check that elements x and y of Co2 are standard generators:
• Check o(x) = 2
• Check o(y) = 5
• Check o(xy) = 28
• Check o(x(xy)14) = 3
This algorithm is available in computer readable format: checker for Co2.

### Presentation

A presentation for Co2 in terms of its standard generators is given below.

< a, b | a2 = b5 = (ab2)9 = [a, b]4 = [a, b2]4 = [a, bab]3 = [a, bab2ab]2 = [a, bab-2]3 = [a, b-2abab-2]2 = (abab2ab-1ab-2)7 = 1 >.

This presentation is available in Magma format as follows: Co2 on a and b.

### Representations

The representations of Co2 available are:

### Maximal subgroups

The maximal subgroups of Co2 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: the linkage 15BC-23AB is compatible with the 748-dimensional representation mod 2. The choice of 14BC is made `without loss of generality'. Go to main ATLAS (version 2.0) page. Go to sporadic groups page. Go to old Co2 page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 26th May 1999.
Last updated 6.1.05 by SJN.
Information checked to Level 1 on 31.05.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.