# ATLAS: Mathieu group M22

Order = 443520.
Mult = 12.
Out = 2.

### Standard generators

Standard generators of the Mathieu group M22 are a and b where a has order 2, b is in class 4A, ab has order 11, and ababb has order 11.
There are problems of 'virtue' in defining standard generators for the various covering groups. The ones defined here may change subtly at a later date.
Standard generators of the double cover 2M22 are pre-images A and B where A is in +2A, B is in -4A, and AB has order 11 (any two of these conditions imply the third).
Standard generators of the triple cover 3M22 are pre-images A and B where A has order 2 and B has order 4.
Standard generators of the fourfold cover 4M22 are pre-images A and B where A has order 2, AB has order 11, and ABABB has order 11.
Standard generators of the sixfold cover 6M22 are pre-images A and B where A is in class +2A, and B is in class -4A.
Standard generators of the twelvefold cover 12M22 are pre-images A and B where A has order 2, B has order 4, AB has order 33, and ABABB has order 33.

Standard generators of the automorphism group M22:2 are c and d where c is in class 2B, d is in class 4C, and cd has order 11.
Standard generators of the double cover 2M22:2 are pre-images C and D where CD has order 11.
Standard generators of the triple cover 3M22:2 are pre-images C and D where CD has order 11. Standard generators of the fourfold cover 4M22:2 are pre-images C and D where CD has order 11. Standard generators of the sixfold cover 6M22:2 are pre-images C and D where CD has order 11. Standard generators of the twelvefold cover 12M22:2 are pre-images C and D where CD has order 11.

### Black box algorithms

To find standard generators for M22:
• Find any element of order 8. Its square is a 4A-element, y, say, and its fourth power is a 2A-element x, say.
• Find a conjugate a of x and a conjugate b of y, whose product has order 11, such that ababb has order 11.
To find standard generators for M22.2:
• Find any element of order 12. Its cube is a 4C-element, y, say.
• Find any element of order 14. Its seventh power is a 2C-element x, say.
• Find a conjugate a of x and a conjugate b of y, whose product has order 11.

### Representations

Representations are available for the following decorations of M22:

#### M22 and covers

The representations of M22 available are
• a and b as permutations on 22 points.
• All faithful irreducibles in characteristic 2.
• a and b as 10 x 10 matrices over GF(2).
• a and b as 10 x 10 matrices over GF(2).
• a and b as 34 x 34 matrices over GF(2).
• a and b as 70 x 70 matrices over GF(4).
• a and b as 70 x 70 matrices over GF(4).
• a and b as 98 x 98 matrices over GF(2).
• All faithful irreducibles in characteristic 3 (up to Frobenius automorphisms).
• a and b as 21 x 21 matrices over GF(3).
• a and b as 45 x 45 matrices over GF(9).
• a and b as 49 x 49 matrices over GF(3).
• a and b as 49 x 49 matrices over GF(3).
• a and b as 55 x 55 matrices over GF(3).
• a and b as 99 x 99 matrices over GF(3).
• a and b as 210 x 210 matrices over GF(3).
• a and b as 231 x 231 matrices over GF(3).
• All faithful irreducibles in characteristic 5 (up to Frobenius automorphisms).
• a and b as 21 x 21 matrices over GF(5).
• a and b as 45 x 45 matrices over GF(25).
• a and b as 55 x 55 matrices over GF(5).
• a and b as 98 x 98 matrices over GF(5).
• a and b as 133 x 133 matrices over GF(5).
• a and b as 210 x 210 matrices over GF(5).
• a and b as 280 x 280 matrices over GF(5).
• a and b as 280 x 280 matrices over GF(5).
• a and b as 385 x 385 matrices over GF(5).
• All faithful irreducibles in characteristic 7.
• a and b as 21 x 21 matrices over GF(7).
• a and b as 45 x 45 matrices over GF(7).
• a and b as 54 x 54 matrices over GF(7).
• a and b as 154 x 154 matrices over GF(7).
• a and b as 210 x 210 matrices over GF(7).
• a and b as 231 x 231 matrices over GF(7).
• a and b as 280 x 280 matrices over GF(49).
• a and b as 280 x 280 matrices over GF(49).
• a and b as 385 x 385 matrices over GF(7).
• All faithful irreducibles in characteristic 11.
• a and b as 20 x 20 matrices over GF(11).
• a and b as 45 x 45 matrices over GF(11).
• a and b as 45 x 45 matrices over GF(11).
• a and b as 55 x 55 matrices over GF(11).
• a and b as 99 x 99 matrices over GF(11).
• a and b as 154 x 154 matrices over GF(11).
• a and b as 190 x 190 matrices over GF(11).
• a and b as 231 x 231 matrices over GF(11).
• a and b as 385 x 385 matrices over GF(11).
The representations of 2.M22 available are
• A and B as 10 x 10 matrices over GF(9).
• A and B as 10 x 10 matrices over GF(25).
• A and B as 120 x 120 matrices over GF(5).
• A and B as 10 x 10 matrices over GF(7).
• A and B as 120 x 120 matrices over GF(7).
• A and B as 126 x 126 matrices over GF(49).
• A and B as 320 x 320 matrices over GF(7).
• A and B as 10 x 10 matrices over GF(11).
The representations of 3.M22 available are
• A and B as 6 x 6 matrices over GF(4).
• A and B as 21 x 21 matrices over GF(25).
• A and B as 21 x 21 matrices over GF(7).
• A and B as 21 x 21 matrices over GF(121).
The representations of 4.M22 available are
• A and B as 56 x 56 matrices over GF(25).
• A and B as 16 x 16 matrices over GF(49).
• A and B as 56 x 56 matrices over GF(121).
The representations of 6.M22 available are
• A and B as 54 x 54 matrices over GF(7).
• A and B as 36 x 36 matrices over GF(121).
The representations of 12.M22 available are
• A and B as 48 x 48 matrices over GF(25).
• A and B as 24 x 24 matrices over GF(121).

#### M22:2 and covers

The representations of M22:2 available are
• c and d as permutations on 22 points.
• c and d as permutations on 77 points.
• c and d as 10 x 10 matrices over GF(2).
The representations of 2.M22:2 available are
• C and D as 10 x 10 matrices over GF(9).
• C and D as 10 x 10 matrices over GF(25).
• C and D as 10 x 10 matrices over GF(7).
• C and D as 10 x 10 matrices over GF(11).
• C and D as 20 x 20 integral matrices, in GAP format (kindly provided by N.J.A.Sloane and G.Nebe from their library of lattices).
The representations of 3.M22:2 available are
• C and D as 12 x 12 matrices over GF(2).
The representations of 4.M22:2 available are
• C and D as 32 x 32 matrices over GF(7).
The representations of 6.M22:2 available are
• C and D as 72 x 72 matrices over GF(11).
The representations of 12.M22:2 available are
• C and D as 48 x 48 matrices over GF(11).

### Maximal subgroups

The maximal subgroups of M22 are as follows. Words provided by Peter Walsh, implemented and checked by Ibrahim Suleiman.
The maximal subgroups of M22:2 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.