ATLAS: Conway group Co_{3}
Order = 495766656000 = 2^{10}.3^{7}.5^{3}.7.11.23.
Mult = 1.
Out = 1.
The following information is available for Co_{3}:
Standard generators of the Conway group Co_{3} are a and
b where a is in class 3A, b is in class 4A and
ab has order 14.
Finding generators
To find standard generators for Co_{3}:
 Find any element of order 9, 18, 24 or 30. It powers up to a 3Aelement x.
 Find any element of order 20. It powers up to a 4Aelement 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 Co_{3}.
Checking generators
To check that elements x and y of Co_{3}
are standard generators:
 Check o(x) = 3
 Check o(y) = 4
 Check o(xy) = 14
 Let t = xyxy^{3}x^{2}
 Let u = (y^{2}(y^{2})^{xyy})^{3}.
 Let v = t(y^{2}(y^{2})^{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 Co_{3}.
The representations of Co_{3} 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:

Dimension 22 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 230 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 896a over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 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:

Dimension 23 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 230 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 253 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 275 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 896b over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducible representations in characteristic 7:

Dimension 23 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 253 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 253 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 275 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 896a over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducible representations in characteristic 11:

Dimension 23 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 253 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 253 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 275 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 896 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducible representations in characteristic 23:

Dimension 23 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 253 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 253 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 274 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 896b over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 23 over Z:
a and b (Magma).
The maximal subgroups of Co_{3} are as follows. Words provided by
Peter Walsh, implemented and checked by Ibrahim Suleiman.

McL:2, with generators
here.

HS, with generators
here.

U4(3).2.2, with generators
here.

M23, with generators
here.

3^5:(2 × M11), with generators
here.

2.S6(2), with generators
here.

U3(5):S3, with generators
here.

3^1+4:4S6, with generators
here.

2^4.A8, with generators
here.

L3(4).D12, with generators
here.

2 × M12, with generators
here.

[2^10.3^3], with generators
here, or (a different copy, with shorter words)
here.

S3 × L2(8):3, with generators
here.

A4 × S5, with generators
here, or (a different copy, with shorter words)
here.
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.