ATLAS: Mathieu group M23

Order = 10200960 =
Mult = 1.
Out = 1.

The following information is available for M23:

Standard generators

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.

Black box algorithms

Finding generators

To find standard generators for M23: 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: 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.

Maximal subgroups

The maximal subgroups of M23 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. These classes are compatible with the Atlas of Brauer Characters.

Checks applied

CheckDateBy whomRemarks
Links work (except representations)12.12.02JNB
Links to (meataxe) representations work and have right degree and field24.01.01RAW
All info from v1 is included24.01.01RAW
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

Main ATLAS page Go to main ATLAS (version 2.0) page.
Sporadic groups page Go to sporadic groups page.
Old M23 page Go to old M23 page - ATLAS version 1.
ftp access 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.