ATLAS: Mathieu group M23
Order = 10200960 = 27.32.5.7.11.23.
Mult = 1.
Out = 1.
The following information is available for M23:
Standard generators of the Mathieu group M23 are a and
b where a has order 2, b has order 4, ab has
order 23 and ababababbababbabb has order 8.
Finding generators
To find standard generators for M23:
- Find any elements x of order 2 and y of order 4.
- Find a conjugate a of x and a conjugate b of y, whose product has order 23,
such that (ab)^2(ababb)^2abb has order 8 or 11. In the latter case, replace b
by its inverse.
This algorithm is available in computer readable format:
finder for M23.
Checking generators
To check that elements x and y of M23
are standard generators:
- Check o(x) = 2
- Check o(y) = 4
- Check o(xy) = 23
- Check o(xyxyxyxyyxyxyyxyy) = 8
This algorithm is available in computer readable format:
checker for M23.
A presentation for M23 in terms of its standard generators is given below.
< a, b | a2 = b4 =
(ab)23 = (ab2)6 =
[a, b]6 =
(abab-1ab2)4 =
(ab)3ab-1ab2(abab-1)2(ab)3(ab-1)3 =
(abab2)3(ab2ab-1)2abab2abab-1ab2
= 1 >.
This presentation is available in Magma format as follows:
M23 on a and b [v1] and
M23 on a and b [v2 - courtesy of Bill Unger].
The representations available are as follows.
They should be in Atlas order, defined by setting
ab in 23B,
abababb in 15A,
abababbb in 7A
and ababababb in 11B.
- All primitive permutation representations.
-
Permutations on 23 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 253 points - the cosets of L3(4).2:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 253 points - the cosets of 2^4.A7:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 506 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 1288 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 1771 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 40320 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- All faithful irreducibles in characteristic 2.
-
Dimension 11 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- the co-code representation.
-
Dimension 11 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- the code representation.
-
Dimension 44 over GF(2):
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).
-
Dimension 120 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 220 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 220 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 252 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 896 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 896 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- Faithful irreducibles in characteristic 3.
-
Dimension 22 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 45 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 104 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 104 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 231 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 253 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 770 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 990 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- Faithful irreducibles in characteristic 5.
-
Dimension 22 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 45 over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 230 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 231 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 231 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 770 over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 896 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 990 over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- Faithful irreducibles in characteristic 7.
-
Dimension 22 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 45 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 208 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 231 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 231 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 231 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 770 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 770 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 896 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 896 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 990 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- Faithful irreducibles in characteristic 11.
-
Dimension 22 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 45 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 229 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 231 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 231 over GF(121):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 231 over GF(121):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 253 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 770 over GF(121):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 770 over GF(121):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 806 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 990 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- Faithful irreducibles in characteristic 23.
-
Dimension 21 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 45 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 210 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 230 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 231 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 253 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 280 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 665b over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The maximal subgroups of M23 are as follows.
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.
These classes are compatible with the Atlas of Brauer Characters.
| Check | Date | By whom | Remarks |
|
| Links work (except representations) | 12.12.02 | JNB | |
| Links to (meataxe) representations work and have right degree and field | 24.01.01 | RAW |
| All info from v1 is included | 24.01.01 | RAW |
| 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 M23 page - ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 7th June 2000.
Last updated 21.12.02 by SJN.
R.A.Wilson, R.A.Parker and J.N.Bray.