ATLAS: Mathieu group M11
Order = 7920 = 24.32.5.11.
Mult = 1.
Out = 1.
The following information is available for M11:
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).
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.
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.
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.
-
Permutations on 11 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 12 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 55 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 66 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 165 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- 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.
-
Dimension 9 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 10 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 10 over GF(11) - the dual of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 11 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 16 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 44 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 55 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- 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.
The maximal subgroups of M11 are as follows.
-
M10 = A6.2, with standard
generators
(ab)^-4a(ab)^4,
(abb)^-2(abababbab)(abb)^2.
To change the previous generators to standard generators of M10, use the program
here.
-
L2(11), with standard generators
a, babbab.
To change the previous generators to standard generators of L2(11), use the program
here.
-
M9:2, with generators
(ab)^-2bab,
(abb)^-1babb.
-
S5 = A5.2, with standard
generators
a, abbabbababbabbab.
To change the previous generators to standard generators of S5, use the program
here.
-
2S4, with generators
a, bababbabab.
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.
| Check | Date | By whom | Remarks |
| Links work (except representations) |
04.02.02 | JNB | |
| Links to (meataxe) representations work and have right degree and field |
24.01.01 | RAW |
| All info from v1 is included | | |
| Valid W3C HTML 4.01 Transitional |
04.02.02 | JNB | This property is very easy to disrupt |
| HTML page standard | | |
| Word program syntax | 24.01.01 | RAW |
| Word programs applied | | |
| All necessary standard generators are defined | 24.01.01 | RAW |
| 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.