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: To find standard generators for M12.2:

Representations

The representations of M12 available are The representations of M12:2 available are The representations of 2.M12 available are The representations of 2.M12:2 available are

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.
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Last updated 13th September 1999,
R.A.Wilson, R.A.Parker and J.N.Bray