ATLAS: Fischer group Fi22

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

The following information is available for Fi22:

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.
Standard generators of the double cover 2.Fi22 are preimages A and B where B has order 13 and AB has order 11.
Standard generators of the triple cover 3.Fi22 are preimages A and B where A has order 2 and B has order 13. The canonical central element is (AB)22

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.
Standard generators of 2.Fi22:2 are preimages C and D where (CD)5D has order 30. This is equivalent to saying D is in +18E and CD is in +42A.
Standard generators of 3.Fi22:2 are preimages C and D where C has order 2.

Black box algorithms

Finding generators

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.
This algorithm is available in computer readable format: finder for 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.
This algorithm is available in computer readable format: finder for Fi22.2.

Checking generators

To check that elements x and y of Fi22 are standard generators:

• Check o(x) = 2
• Check o(y) = 13
• Check o(xy) = 11
• Check o(xyxyxyxyyxyxyyxyy) = 12
• Let z = xyxyyxyy
• Check o(z) = 30
• Check o(xz15) = 3
This algorithm is available in computer readable format: checker for Fi22.

To check that elements x and y of Fi22.2 are standard generators:

• Check o(x) = 2
• Check o(y) = 18
• Check o(xy) = 42
• Let z = xyxy5xy4
• Check o(z) = 22
• Check o(xz11) = 3
• Check o((y9)xyyy(xy)21) = 3
This algorithm is available in computer readable format: checker for Fi22.2.

Presentations

Presentations of Fi22 and Fi22:2 in terms of their standard generators are given below. (The relation (cd9)4 = 1 in the Fi22:2 presentation is redundant.)

< a, b | a2 = b13 = (ab)11 = (ab2)21 = [a, b]3 = [a, b2]3 = [a, b3]3 = [a, b4]2 = [a, b5]3 = [a, bab2]3 = [a, b-1ab-2]2 = [a, bab5]2 = [a, b2ab5]2 = 1 >.

< c, d | c2 = d18 = [c, d]3 = [c, d2]3 = [c, d3]3 = [c, d4]3 = [c, d5]3 = [c, d6]2 = [c, d7]2 = [c, d8]3 = (cd9)4 = [cdcdcd-2cd] = [cd2cd2cd-4cd2] = ((cd3)4cd-4)2 = (cd4cd5cd5)5 = (cdcd-3)8 = 1 >.

These presentations are available in Magma format as follows:
Fi22 on a and b, 2.Fi22 on A and B, Fi22:2 on c and d and 3.Fi22:2 on C and D.

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:
• Permutations on 3510 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 14080 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 61776 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 142155 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 694980 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some irreducible representations over GF(2):
• Some irreducible representations over GF(3):
• Dimension 77 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 351 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 924 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Also available in dimension 4823 over GF(3) - send email for details.
• Dimension 78 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 428 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 78 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 429 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 78 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 429 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 78 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 429 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
The representations of 2.Fi22 available are:
The representations of 3.Fi22 available are:
The representations of 6.Fi22 available are:
• Permutations on 370656 points - kindly provided by Bernd Schröder. A and B (GAP).
The representations of Fi22:2 available are:
• Permutations on 3510 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 78 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 350 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 572 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 1352 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 77 over GF(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 351 over GF(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 924 over GF(3) (6/10/04 - corrected to standard generators): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 78 over GF(5): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 428 over GF(5): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 78 over GF(7): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 429 over GF(7): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 78 over GF(11): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 429 over GF(11): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 78 over GF(13): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 429 over GF(13): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 78 over Z - kindly provided by Bernd Schröder. c and d (GAP).
The representations of 2.Fi22:2 available are:
The representations of 2.Fi22.4 (a group of shape 2.Fi22.4, in which outer `involutions' square to a scalar of order 4) available are:
The representations of 3.Fi22:2 available are:
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 of Fi22 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.

A set of generators for the maximal cyclic subgroups of Fi22:2 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. Go to main ATLAS (version 2.0) page. Go to sporadic groups page. Go to old Fi22 page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 17th January 2001.
Last updated 3.8.06 by RWB.
Information checked to Level 0 on 17.01.01 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.