Order = 10200960 = 27.32.5.7.11.23.
Mult = 1.
Out = 1.

## Porting notes

Porting incomplete.

## Standard generators

Standard generators of M23 are a, b where a has order 2, b has order 4, ab has order 23 and ababababbababbabb has order 8.

## Black box algorithms

### Finding generators

Group Algorithm File

### Checking generators (semi-presentations)

Group Semi-presentation File
M23 〈〈 a, b | o(a) = 2, o(b) = 4, o(ab) = 23, o(abababab2abab2ab2) = 8 〉〉 Download

## Presentations

M23 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 〉 Details
M23 a, b | a2 = b4 = (ab2)6 = (abab−1ab2)4 = abababab−1ab2abab−1abab−1abababab−1ab−1ab−1 = abab2abab2abab2ab2ab−1ab2ab−1abab2abab−1ab2 = abab2ab2abab2ab2abab2ab2abab2ab2ab−1ab2ab2ab−1ab2ab2 = ababababab2abab−1abab2ababab−1ab2abab2ab2abab−1ab−1abab2 = 1 〉 Details

## Maximal subgroups

### Maximal subgroups of M23

Subgroup Order Index Programs/reps
M22 443 520 23 Program: Standard generators
L3(4):22 40 320 253 Program: Generators
24:A7 40 320 253 Program: Generators
A8 20 160 506 Program: Standard generators
M11 7 920 1 288 Program: Standard generators
24:(3 × A5):2 5 760 1 771 Program: Generators
23:11 253 40 320 Program: Generators

## Conjugacy classes

### Conjugacy classes of M23

Conjugacy class Centraliser order Power up Class rep(s)
1A 10 200 960 ababbababbababbababbababbababbababbababb
2A 2 688 4A 6A 8A 14A 14B ababbababbababbababb
3A 180 6A 15A 15B abbabb
4A 32 8A ababbababb
5A 15 15A 15B (abababb)3
6A 12 abb
7A 14 7B3 14A 14B (bababababbabbbababababbabb)3
7B 14 7A3 14A 14B bababababbabbbababababbabb
8A 8 ababb
11A 11 11B2 ababababbababababb
11B 11 11A2 ababababb
14A 14 14B3 (bababababbabb)3
14B 14 14A3 bababababbabb
15A 15 15B7 abababb
15B 15 15A7 (abababb)7
23A 23 23B5 (ab)5
23B 23 23A5 ab