# ATLAS: Conway group Co3

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

The following information is available for Co3:

### 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

#### Finding generators

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.
This algorithm is available in computer readable format: finder for Co3.

#### Checking generators

To check that elements x and y of Co3 are standard generators:
• Check o(x) = 3
• Check o(y) = 4
• Check o(xy) = 14
• Let t = xyxy3x2
• Let u = (y2(y2)xyy)3.
• Let v = t(y2(y2)t)2
• Let w = (uvv)3(uv)6
• Check o(w) = 5
• Check o([w,y]) = 1
This algorithm is available in computer readable format: checker for Co3.

### Representations

The representations of Co3 available are:
• Some permutation representations:
• Permutations on 276 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 552 points - imprimitive: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 11178 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 37950 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 48600 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 128800 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some irreducible representations in characteristic 2:
• Some irreducible representations in characteristic 3:
• Dimension 22 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 126 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 126 over GF(3) - the dual of the above.: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
[NB: The ordering of the representations of degree 126 over GF(3) has been changed from version 1.]
• Dimension 231 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 231 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 770 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 770 over GF(3) - the dual of the above.: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some irreducible representations in characteristic 5:
• Some irreducible representations in characteristic 7:
• Some irreducible representations in characteristic 11:
• Some irreducible representations in characteristic 23:
• Dimension 23 over Z: a and b (Magma).

### 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 have been dealt with - the class names are consistent with the mod 3 character table in GAP. Go to main ATLAS (version 2.0) page. Go to sporadic groups page. Go to old Co3 page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 14th June 2000.
Last updated 6.1.05 by SJN.
Information checked to Level 0 on 14.06.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.