ATLAS: Mathieu group M_{12}
Order = 95040 = 2^{6}.3^{3}.5.11.
Mult = 2.
Out = 2.
The following information is available for M_{12}:
Standard generators
Standard generators of M_{12} 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.M_{12} 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 M_{12}: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.M_{12}.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 M_{12} 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 M_{12} are:
- M11, with standard generators
a,
(abb)^-3ababababbababbabb(abb)^3.
- M11, with standard generators
(ab)^-2a(ab)^2,
(abb)^-2ababababbabbababb(abb)^2.
- A6.2.2, with generators
(bab)^-1abab,
(abb)^-1ababababbabbababbabb
.
- A6.2.2, with generators
(ab)^-3a(ab)^3,
(abb)^-2ab(ababb)^2abb(abb)^2.
- L2(11), with standard generators
(abbabab)^3,
(abb)^-2b(abb)^2.
- 3^2:2S4, with generators
here.
- 3^2:2S4, with generators
here.
- 2 × S5, with generators (mapping to standard generators of S5)
(ab)^-1(abababb)^3ab,
(abb)^-3(abababbababbabb)(abb)^3.
- 2^1+4:S3, with generators
here, or here.
- 4^2:D12, with generators
here.
- A4 × S3, with generators
(ab)^-1(abababb)^3ab,
(abb)^-1babb.
The maximal subgroups of M_{12}: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