ATLAS: Mathieu group M_{11}
Order = 7920 = 2^{4}.3^{2}.5.11.
Mult = 1.
Out = 1.
The following information is available for M_{11}:
Standard generators
Standard generators of M_{11} are a and b where
a has order 2, b has order 4, ab has order 11 and
ababababbababbabb has order 4. Two equivalent conditions to the last one
are that ababbabbb has order 5 or that ababbbabb has order 3.
In the natural representation we may take
a = (2, 10)(4, 11)(5, 7)(8, 9) and
b = (1, 4, 3, 8)(2, 5, 6, 9).
Black box algorithms
To find standard generators for M_{11}:
- Find an element of order 4 or 8. This powers up to x of order 2 and y of order 4.
[The probability of success at each attempt is 3 in 8 (about 1 in 3).]
- Find a conjugate a of x and a conjugate b of y such that ab has order 11.
[The probability of success at each attempt is 16 in 165 (about 1 in 10).]
- If ababbabbb has order 3, then replace b by its inverse.
- Now ababbabbb has order 5, and standard generators of M_{11}
have been obtained.
Presentation
A presentation for M_{11} in terms of its standard generators is given below.
< a, b | a^{2} = b^{4} = (ab)^{11} = (ab^{2})^{6} = ababab^{-1}abab^{2}ab^{-1}abab^{-1}ab^{-1} = 1 >.
Representations
The representations available are as follows. They should follow the order
in the ATLAS of Brauer Characters, with the conjugacy classes
defined by ab in 11A and ababababb in 8B,
but please check this yourself if you rely on it!
- All primitive permutation representations.
- a and
b as
the above permutations on 11 points.
- a and
b as
permutations on 12 points.
- a and
b as
permutations on 55 points.
- a and
b as
permutations on 66 points.
- a and
b as
permutations on 165 points.
- 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
32 × 32 matrices over GF(2) - reducible over GF(4).
- a and
b as
44 × 44 matrices over GF(2).
- All faithful irreducibles in characteristic 3.
- a and
b as
5 × 5 matrices over GF(3) - the cocode representation.
- a and
b as
5 × 5 matrices over GF(3) - the code representation.
- a and
b as
10 × 10 matrices over GF(3) - the deleted permutation representation.
- a and
b as
10 × 10 matrices over GF(3) - the skewsquare of the code representation.
- a and
b as
10 × 10 matrices over GF(3) - the skewsquare of the cocode representation.
- a and
b as
24 × 24 matrices over GF(3).
- a and
b as
45 × 45 matrices over GF(3).
- All faithful irreducibles in characteristic 5.
- a and
b as
10 × 10 matrices over GF(5).
- a and
b as
10 × 10 matrices over GF(25).
- a and
b as
10 × 10 matrices over GF(25) - the dual of the above.
- a and
b as
11 × 11 matrices over GF(5).
- a and
b as
16 × 16 matrices over GF(5).
- a and
b as
16 × 16 matrices over GF(5) - the dual of the above.
- a and
b as
20 × 20 matrices over GF(5) - reducible over GF(25).
- a and
b as
45 × 45 matrices over GF(5).
- a and
b as
55 × 55 matrices over GF(5).
- All faithful irreducibles in characteristic 11.
- a and
b as
9 × 9 matrices over GF(11).
- a and
b as
10 × 10 matrices over GF(11).
- a and
b as
10 × 10 matrices over GF(11) - the dual of the above.
- a and
b as
11 × 11 matrices over GF(11).
- a and
b as
16 × 16 matrices over GF(11).
- a and
b as
44 × 44 matrices over GF(11).
- a and
b as
55 × 55 matrices over GF(11).
- Some faithful irreducibles in characteristic 0.
- a and b as
10 × 10 matrices over Z.
- a and b as
10 × 10 matrices over Z[i2].
- a and b as
10 × 10 matrices over Z[i2] - the dual of the above.
- a and b as
11 × 11 monomial matrices over Z.
- a and b as
20 × 20 matrices over Z - reducible over Q(i2).
- a and b as
32 × 32 matrices over Z - reducible over Q(b11).
- a and b as
44 × 44 matrices over Z.
- a and b as
45 × 45 matrices over Z.
- a and b as
55 × 55 monomial matrices over Z.
Sources: All the above representations, except those in characteristic 0, are
easily obtained with the Meataxe from the permutation representations on 11 and
12 points. Most of the representations in characteristic 0 are not that
difficult to obtain either (the most difficult being the nonrational
representations of degree 10).
NB: There is some ambiguity as to which of the two 5dimensional GF(3)modules of M_{11} should be regarded as the code and which
as the cocode. Let M = 2M_{12} be the full automorphism group
of the ternary Golay code. So M monomially permutes the vectors
e_{1}, e_{2}, . . . , e_{12} (and their negatives). Now M has two conjugacy classes of subgroups isomorphic to M_{11} and their representatives may be taken to be M_{1}, stabilising e_{1}, and M_{2}, the subgroup of (pure) permutations. The terms `code' and `cocode' used above refer to M_{1} and NOT to M_{2}.
In the GF(3)representation 5a, M_{11} has orbits 11 + 110 on points and orbits 22 + 220 on nonzero vectors.
In the GF(3)representation 5b, M_{11} has orbits 55 + 66 on points and orbits 132 + 110 on nonzero vectors.
Maximal subgroups
The maximal subgroups of M_{11} are as follows.
- M_{10} = A_{6}.2, with standard
generators (ab)^-4a(ab)^4,
(abb)^-2(abababbab)(abb)^2.
- L_{2}(11), with standard generators
a, babbab.
- M_{9}:2, with generators (ba)^-1aba,
(ab)^-1bab or (ab)^-2bab,
(abb)^-1babb.
- S_{5} = A_{5}.2, with standard
generators a, abbabbababbabbab.
- 2S_{4}, with generators a, bababbabab.
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 have been dealt with.
Representatives of the 10 conjugacy classes of M_{11} are also given below.
- 1A: identity [or a^{2}].
- 2A: a.^{ }
- 3A: ab^{2}ab^{2}.
- 4A: b.^{ }
- 5A: abab^{2}ab^{-1}.
- 6A: ab^{2}.
- 8A: abab^{2}ab^{2}.
- 8B: ab^{-1}ab^{2}ab^{2}.
- 11A: ab.^{ }
- 11B: ab^{-1}.
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Last updated 20th August 1999,
R.A.Wilson, R.A.Parker and J.N.Bray