ATLAS: Mathieu group M_{23}
Order = 10200960 = 2^{7}.3^{2}.5.7.11.23.
Mult = 1.
Out = 1.
The following information is available for M_{23}:
Standard generators of the Mathieu group M_{23} 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 M_{23}:
 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 M_{23}.
Checking generators
To check that elements x and y of M_{23}
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 M_{23}.
A presentation for M_{23} in terms of its standard generators is given below.
< a, b  a^{2} = b^{4} =
(ab)^{23} = (ab^{2})^{6} =
[a, b]^{6} =
(abab^{1}ab^{2})^{4} =
(ab)^{3}ab^{1}ab^{2}(abab^{1})^{2}(ab)^{3}(ab^{1})^{3} =
(abab^{2})^{3}(ab^{2}ab^{1})^{2}abab^{2}abab^{1}ab^{2}
= 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 cocode 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 M_{23} 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.