# ATLAS: Conway group Co1

Order = 4157776806543360000 = 221.39.54.72.11.13.23.
Mult = 2.
Out = 1.

The following information is available for Co1:

### 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 2.Co1 = Co0 are preimages A and B where B has order 3 and ABABABABBABABBABB has order 33.

### Black box algorithms

#### Finding generators

To find standard generators for Co1:

• Find any element of order 26 or 42. It powers up to a 2B­element x.
[The probability of success at each attempt is 17 in 546 (about 1 in 32).]
• Find any element of order divisible by 9 (i.e. 9, 18 or 36). It powers up to a 3C­element y.
[The probability of success at each attempt is 2 in 27 (about 1 in 14).]
• Find a conjugate a of x and a conjugate b of y such that ab has order 40 and ababb has order 6.
[The probability of success at each attempt is 54 in 8855 (about 1 in 164).]
• Now a and b are standard generators of Co1.
This algorithm is available in computer readable format: finder for Co1.

#### Checking generators

To check that elements x and y of Co1 are standard generators:
• Check o(x) = 2
• Check o(y) = 3
• Check o(xy) = 40
• Check o(xyxyy) = 6
• Let z = xy(xyxyy)2
• Check o(z) = 42
• Check o(xyyz21) = 11
• Let u = (xy)20
• Let v = yxyxyxyxyy
• Check o(uv) = 36
• Check o(uvvuvuv) = 18
This algorithm is available in computer readable format: checker for Co1.

### Representations

The representations of Co1 available are:
The representations of 2.Co1 available are:

### Maximal subgroups

The maximal subgroups of Co1 are as follows.

### 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.
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