# ATLAS: Mathieu group M12

Order = 95040 = 26.33.5.11.
Mult = 2.
Out = 2.

The following information is available for M12:

### Standard generators

Standard generators of M12 are a and b where a is in class 2B, b is in class 3B and ab has order 11.
Standard generators of the double cover 2.M12 are preimages A and B where A is in class +2B, B has order 6 and AB has order 11. (Note that any two of these conditions imply the third.)

Standard generators of M12:2 are c and d where c is in class 2C, d is in class 3A and cd is in class 12A. (This last condition can be replaced by: cd has order 12 and cdcdd has order 11.)
Standard generators of either of the double covers 2.M12.2 are preimages C and D where D has order 3.

A pair of elements automorphic to A, B can be obtained as
A' = (CDCDCDDCD)3, B' = (CDD)-3(CD)4(CDD)3.

### Black box algorithms

To find standard generators for M12:
• Find any element of order 8. Its fourth power is a 2B-element x, say.
• Find any 3-element y, say.
• Find a conjugate a of x and a conjugate b of y, whose product has order 11. If ababb has order 6, then these are standard generators of M12. Otherwise (when ababb has order 5), the 3-element is in the wrong conjugacy class, so try again.
To find standard generators for M12.2:
• Find any element of order 10. Its fifth power is a 2C-element x, say.
• etc.

### Representations

The representations of M12 available are
• All faithful irreducibles in characteristic 2.
• a and b as 10 × 10 matrices over GF(2).
• a and b as 16 × 16 matrices over GF(4).
• a and b as 16 × 16 matrices over GF(4) - the dual of the above.
• a and b as 44 × 44 matrices over GF(2).
• a and b as 144 × 144 matrices over GF(2).
• All faithful irreducibles in characteristic 3 (up to automorphisms).
• a and b as 10 × 10 matrices over GF(3).
• a and b as 15 × 15 matrices over GF(3).
• a and b as 34 × 34 matrices over GF(3).
• a and b as 45 × 45 matrices over GF(3).
• a and b as 45 × 45 matrices over GF(3).
• a and b as 54 × 54 matrices over GF(3).
• a and b as 99 × 99 matrices over GF(3).
• All faithful irreducibles in characteristic 5 (up to automorphisms).
• a and b as 11 × 11 matrices over GF(5).
• a and b as 16 × 16 matrices over GF(5).
• a and b as 45 × 45 matrices over GF(5).
• a and b as 55 × 55 matrices over GF(5).
• a and b as 55 × 55 matrices over GF(5).
• a and b as 66 × 66 matrices over GF(5).
• a and b as 78 × 78 matrices over GF(5).
• a and b as 98 × 98 matrices over GF(5).
• a and b as 120 × 120 matrices over GF(5).
• All faithful irreducibles in characteristic 11 (up to automorphisms).
• a and b as 11 × 11 matrices over GF(11).
• a and b as 16 × 16 matrices over GF(11).
• a and b as 29 × 29 matrices over GF(11).
• a and b as 53 × 53 matrices over GF(11).
• a and b as 55 × 55 matrices over GF(11).
• a and b as 55 × 55 matrices over GF(11).
• a and b as 66 × 66 matrices over GF(11).
• a and b as 91 × 91 matrices over GF(11).
• a and b as 99 × 99 matrices over GF(11).
• a and b as 176 × 176 matrices over GF(11).
• Permutation representations.
• a and b as permutations on 12 points.
• a and b as permutations on 12 points - the image of the above under an outer automorphism.
• a and b as permutations on 66 points.
• a and b as permutations on 66 points - the image of the above under an outer automorphism.
The representations of M12:2 available are
• All 2-modular irreducibles.
• c and d as 10 × 10 matrices over GF(2).
• c and d as 32 × 32 matrices over GF(2).
• c and d as 44 × 44 matrices over GF(2).
• c and d as 144 × 144 matrices over GF(2).
• c and d as permutations on 24 points.
The representations of 2.M12 available are
• A and B as permutations on 24 points.
• A and B as 6 × 6 matrices over GF(3).
• A and B as 12 × 12 matrices over GF(5).
The representations of 2.M12:2 available are
• C and D as 10 × 10 matrices over GF(3).
• C and D as 12 × 12 matrices over GF(3).
• C and D as permutations on 48 points.

### Maximal subgroups

The maximal subgroups of M12 are:
The maximal subgroups of M12:2 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. There are no problems of algebraic conjugacy.