# ATLAS: Fischer group Fi23

Order = 4089470473293004800 = 218.313.52.7.11.13.17.23.
Mult = 1.
Out = 1.

The following information is available for Fi23:

### Standard generators

Standard generators of the Fischer group Fi23 are a and b where a is in class 2B, b is in class 3D and ab has order 28.

### Black box algorithms

#### Finding generators

To find standard generators for Fi23:
• Find any element of order 20, 28 or 60. It powers up to a 2B-element x.
• Find any element of order 3, y say, (probably as a power of an element of order 9 or 18).
• Find a conjugate a of x and a conjugate b of y whose product has order 28.
• If you succeed, standard generators for Fi23 have been obtained.
• If you fail, then y is probably in the wrong conjugacy class.
This algorithm is available in computer readable format: finder for Fi23.

#### Checking generators

To check that elements x and y of Fi23 are standard generators:

• Check o(x) = 2
• Check o(y) = 3
• Check o(xy) = 28
• Check o(xyy(xy)14) = 5
This algorithm is available in computer readable format: checker for Fi23.

### Representations

The representations of Fi23 available are:

### Maximal subgroups

The maximal subgroups of Fi23 are:

### 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.
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