# ATLAS: Fischer group Fi24'

Order = 1255205709190661721292800 = 221.316.52.73.11.13.17.23.29.
Mult = 3.
Out = 2.

The following information is available for Fi24':

### Standard generators

Standard generators of the Fischer group Fi24' are a and b where a is in class 2A, b is in class 3E, ab has order 29 and abababb has order 33.
Standard generators of the triple cover 3.Fi24' are preimages A and B where A has order 2 and AB has order 29.

Standard generators of the automorphism group Fi24 = Fi24':2 are c and d where c is in class 2C, d is in class 8D and cd has order 29.
Standard generators of 3.Fi24':2 are preimages C and D where D has order 29.

A pair of generators conjugate to a, b can be obtained as
a' = (cdd)10, b' = (cdcdd)−10(cdcdcddcdcdd)8(cdcdd)10.

### Black box algorithms

#### Finding generators

To find standard generators for Fi24':

• Find an element of order 22, 26, 28 or 60. It powers up to a 2A-element x.
• Find an element of order 39. It has a 50-50 chance of powering up to a 3E-element y.
• Find conjugates a of x and b of y whose product has order 29 [probability about 1 in 127].
• If you fail, you probably have a 3A-element, so go back to step 2.
• If you succeed, you definitely have a 3E-element, so keep going until abababb has order 33.
This algorithm is available in computer readable format: finder for Fi24'.

To find standard generators of Fi24 = Fi24':2:

• Find an element of order 34, 46, 54, 70, or 78. It powers to an element x in class 2C.
• Find an element of order 40. It powers to an element y in class 8D.
• Find conjugates c of x and d of y such that cd has order 29. These are standard generators.
This algorithm is available in computer readable format: finder for Fi24':2.

#### Checking generators

To check that elements x and y of Fi24' are standard generators:

• Check o(x) = 2
• Check o(y) = 3
• Check o(xy) = 29
• Check o(xyxyxyy) = 23
• Let z = (xy)6y
• Check o(z) = 60
• Check o(x(z30)xyxy) = 5
This algorithm is available in computer readable format: checker for Fi24'.

To check that elements x and y of Fi24':2 are standard generators:

• Check o(x) = 2
• Check o(y) = 8
• Check o(xy) = 29
• Let z = xyxy6
• Check o(z) = 54
• Check o(xz27) = 3
• Check o(xyy) = 20
This algorithm is available in computer readable format: checker for Fi24':2.

### Representations

The representations of Fi24' available are:
The representations of 3.Fi24' available are:
The representations of Fi24 = Fi24':2 available are:
The representations of 3.Fi24':2 available are:

### Maximal subgroups

The maximal subgroups of the simple group Fi24' are:
The maximal subgroups of the automorphism group Fi24 = Fi24':2 are:

### Conjugacy classes

A set of generators for the maximal cyclic subgroups up to automorphism can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements, or their images under an outer automorphism. Problems of algebraic conjugacy are also not dealt with. Go to main ATLAS (version 2.0) page. Go to sporadic groups page. Go to old Fi24' page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 16th June 2000.
Last updated 16.05.06 by JNB.
Information checked to Level 0 on 16.06.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.