# 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

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:
• 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.

### Presentation

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].

### Representations

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.
• Faithful irreducibles in characteristic 5.
• Faithful irreducibles in characteristic 7.
• Faithful irreducibles in characteristic 11.
• Faithful irreducibles in characteristic 23.

### 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 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

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