# ATLAS: Mathieu group M11

Order = 7920 = 24.32.5.11.
Mult = 1.
Out = 1.

The following information is available for M11:

### Standard generators

Standard generators of M11 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 M11:
• 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 M11 have been obtained.

### Presentation

A presentation for M11 in terms of its standard generators is given below.

< a, b | a2 = b4 = (ab)11 = (ab2)6 = ababab-1abab2ab-1abab-1ab-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 M11 should be regarded as the code and which as the cocode. Let M = 2M12 be the full automorphism group of the ternary Golay code. So M monomially permutes the vectors e1, e2, . . . , e12 (and their negatives). Now M has two conjugacy classes of subgroups isomorphic to M11 and their representatives may be taken to be M1, stabilising e1, and M2, the subgroup of (pure) permutations. The terms `code' and `cocode' used above refer to M1 and NOT to M2.

In the GF(3)­representation 5a, M11 has orbits 11 + 110 on points and orbits 22 + 220 on nonzero vectors.
In the GF(3)­representation 5b, M11 has orbits 55 + 66 on points and orbits 132 + 110 on nonzero vectors.

### Maximal subgroups

The maximal subgroups of M11 are as follows.

### 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 M11 are also given below.

• 1A: identity [or a2].
• 2A: a.
• 3A: ab2ab2.
• 4A: b.
• 5A: abab2ab-1.
• 6A: ab2.
• 8A: abab2ab2.
• 8B: ab-1ab2ab2.
• 11A: ab.
• 11B: ab-1.