# ATLAS: Fischer group Fi22

Order = 64561751654400 = 217.39.52.7.11.13.
Mult = 6.
Out = 2.

### Standard generators and automorphisms

Standard generators of the Fischer group Fi22 are a and b where a is in class 2A, b has order 13, ab has order 11 and ababababbababbabb has order 12.

The outer automorphism may be realised by mapping (a, b) to a, (ab)^4bb(ab)^-4.

Standard generators of the automorphism group Fi22:2 are c and d where c is in class 2A, d is in class 18E and cd has order 42.

### Black box algorithms

To find standard generators for Fi22:
• Find any element of order 14, 22 or 30. This then powers to a 2A-element x.
• Find any element of order 13, y, say.
• Find a conjugate a of x and a conjugate b of y, whose product has order 11, and such that (ab)2(ababb)2abb has order 12. These are standard generators of Fi22.
To find standard generators for Fi22.2:
• Find any element of order 22. Its 11th power is a 2A-element x, say.
• Find an element y of order 42.
• Find a conjugate c of x and a conjugate z of y such that cz has order 18 and cz5 has order 42.
• Now c and d = cz are standard generators of Fi22:2.

### Representations

Representations are available for groups isoclinic to one of the following:
[Actually, representations of 6.Fi22:2 are not yet available.]

The representations of Fi22 available are

• Some primitive permutation representations:
• a and b as permutations on 3510 points.
• a and b as permutations on 14080 points - second of the 2 reps of this degree wrt the words Fi22-O73 below.
• a and b as permutations on 61776 points.
• a and b as permutations on 142155 points.
• a and b as permutations on 694980 points.
• Some irreducible representations over GF(2):
• a and b as 78 × 78 matrices over GF(2).
• a and b as 350 × 350 matrices over GF(2).
• a and b as 572 × 572 matrices over GF(2).
• Some irreducible representations over GF(3):
• a and b as 77 × 77 matrices over GF(3).
• a and b as 351 × 351 matrices over GF(3).
• a and b as 924 × 924 matrices over GF(3).
• Also available as 4823 × 4823 matrices over GF(3) - send email for details.
• a and b as 78 × 78 matrices over GF(5).
• a and b as 78 × 78 matrices over GF(7).
• a and b as 78 × 78 matrices over GF(11).
• a and b as 78 × 78 matrices over GF(13).
The representations of 2.Fi22 available are
• A and B as 176 × 176 matrices over GF(3).
• A and B as 352 × 352 matrices over GF(5).
• A and B as permutations on 28160 points.
The representations of 3.Fi22 available are
• A and B as 27 × 27 matrices over GF(4).
• A and B as 351 × 351 matrices over GF(7) - kindly provided by John Bray.
The representations of 6.Fi22 available are
• A and B as permutations on 370656 points - kindly provided by Bernd Schröder.
The representations of Fi22:2 available are
• c and d as permutations on 3510 points.
• c and d as 78 × 78 matrices over GF(2).
• c and d as 77 × 77 matrices over GF(3).
• c and d as 78 × 78 matrices over Z - kindly provided by Bernd Schröder.
The representations of 2.Fi22:2 available are
• C and D as 352 × 352 matrices over GF(3).
• C and D as 352 × 352 matrices over GF(5). This is actually an isoclinic group of shape 2.Fi22.4, in which outer `involutions' square to a scalar of order 4.
The representations of 3.Fi22:2 available are
• C and D as 54 × 54 matrices over GF(2).
• C and D as 702 × 702 matrices over GF(7).
• C and D as permutations on 185328 points.
The representations of 6.Fi22:2 available are
• none

### Maximal subgroups

The maximal subgroups of Fi22 are
The maximal subgroups of Fi22:2 are
We add here some subgroups which may be useful for condensation purposes:

### 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.

R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk
jnb@for.mat.bham.ac.uk