# 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

#### Finding generators

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.
This algorithm is available in computer readable format: finder for M11.

#### Checking generators

To check that elements x and y of M11 are standard generators:
• Check o(x) = 2
• Check o(y) = 4
• Check o(xy) = 11
• Check o(xyxyyxyyy) = 5
This algorithm is available in computer readable format: checker for M11.

### 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 >.

This presentation is available in Magma format as follows: M11 on a and b.

### 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.
• All faithful irreducibles in characteristic 2.
• Dimension 10 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 16 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 16 over GF(4) - the dual of the above: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 32 over GF(2) - reducible over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 44 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• All faithful irreducibles in characteristic 3.
• Dimension 5 over GF(3) - the cocode representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 5 over GF(3) - the code representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 10 over GF(3) - the deleted permutation representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 10 over GF(3) - the skew square of the code representation:
a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 10 over GF(3) - the skew square of the cocode representation:
a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 24 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 45 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• All faithful irreducibles in characteristic 5.
• Dimension 10 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 10 over GF(25): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 10 over GF(25) - the dual of the above: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 11 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 16 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 16 over GF(5) - the dual of the above: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 20 over GF(5) - reducible over GF(25): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 45 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 55 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• All faithful irreducibles in characteristic 11.
• Some faithful irreducibles in characteristic 0.
• Dimension 10 over Z: a and b (Magma).
• Dimension 10 over Z[i2]: a and b (Magma).
• Dimension 10 over Z[i2] - the complex conjugate: a and b (Magma).
• Dimension 11 over Z: a and b (Magma).
• Dimension 20 over Z - reducible over Q(i2): a and b (Magma).
• Dimension 32 over Z - reducible over Q(b11): a and b (Magma).
• Dimension 44 over Z: a and b (Magma).
• Dimension 45 over Z: a and b (Magma).
• Dimension 55 over Z: a and b (Magma).

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, for example by running this program afterwards. All conjugacy classes can be obtained directly from the standard generators by running this program.

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.

### Checks applied

CheckDateBy whomRemarks
Links work (except representations) 04.02.02JNB
Links to (meataxe) representations work and have right degree and field 24.01.01RAW
All info from v1 is included
Valid W3C HTML 4.01 Transitional 04.02.02JNBThis property is very easy to disrupt
HTML page standard
Word program syntax24.01.01RAW
Word programs applied
All necessary standard generators are defined24.01.01RAW
All representations are in standard generators Go to main ATLAS (version 2.0) page. Go to sporadic groups page. Go to old M11 page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 13th April 1999.
Last updated 21.12.04 by SJN.
Information checked to Level 1 on 03.12.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.