Order = 244823040 = 210.33.5.7.11.23.
Mult = 1.
Out = 1.
Porting notes
Porting incomplete.Standard generators
Standard generators of M24 are a, b where a is in class 2B, b is in class 3A, ab has order 23 and abababbababbabb has order 4.
Black box algorithms
Finding generators
| Group | Algorithm | File |
|---|---|---|
| M24 |
| Download |
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| M24 | 〈〈 a, b | o(a) = 2, o(b) = 3, o(ab) = 23, o(abababbababbabb) = 4, o(abababb) = 12 〉〉 | Download |
Presentations
| Group | Presentation | Link |
|---|---|---|
| M24 | 〈 a, b | a2 = b3 = (ab)23 = [a, b]12 = [a, bab]5 = (ababab−1)3(abab−1ab−1)3 = (ab(abab−1)3)4 = 1 〉 | Details |
Representations
Representations of M24
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 24 Std Details 276 Std Details 759 Std Details 1288 Std Details 1771 Std Details 2024 Std Details 3795 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z 23 Std Details 0 Z[b7, ½] 45 a Std Details 0 Z[b7, ½] 45 b Std Details 0 Q(b15) 231 a Std Details 0 Q(b15) 231 b Std Details Char Ring Dimension ID Generators Description Link 2 GF(2) 11 a Std The Golay code Details 2 GF(2) 11 b Std The Golay cocode Details 2 GF(2) 44 a Std Details 2 GF(2) 44 b Std Details 2 GF(2) 120 Std Details 2 GF(2) 220 a Std Details 2 GF(2) 220 b Std Details 2 GF(2) 252 Std Details 2 GF(2) 320 a Std Details 2 GF(2) 320 b Std Details 2 GF(2) 1242 Std Details 2 GF(2) 1792 Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 22 Std Details 3 GF(9) 45 a Std Details 3 GF(9) 45 b Std Details 3 GF(3) 231 Std Details 3 GF(3) 252 Std Details 3 GF(3) 483 Std Details 3 GF(3) 770 a Std Details 3 GF(3) 770 b Std Details 3 GF(9) 990 b Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 23 Std Details 5 GF(25) 45 a Std Details 5 GF(25) 45 b Std Details 5 GF(5) 231 Std Details 5 GF(5) 252 Std Details 5 GF(5) 253 Std Details 5 GF(25) 770 a Std Details 5 GF(25) 990 a Std Details Char Ring Dimension ID Generators Description Link 7 GF(7) 23 Std Details 7 GF(7) 45 Std Details 7 GF(49) 231 b Std Details 7 GF(7) 252 Std Details 7 GF(7) 253 Std Details 7 GF(7) 483 Std Details 7 GF(49) 770 a Std Details 7 GF(7) 990 Std Details Char Ring Dimension ID Generators Description Link 11 GF(11) 23 Std Details 11 GF(11) 45 Std Details 11 GF(11) 229 Std Details 11 GF(121) 231 b Std Details 11 GF(11) 253 Std Details 11 GF(11) 482 Std Details 11 GF(121) 770 a Std Details 11 GF(11) 806 Std Details 11 GF(11) 990 b Std Details Char Ring Dimension ID Generators Description Link 23 GF(23) 23 Std Details 23 GF(23) 45 Std Details 23 GF(23) 231 b Std Details 23 GF(23) 251 Std Details 23 GF(23) 253 Std Details 23 GF(23) 483 Std Details 23 GF(23) 770 Std Details 23 GF(23) 990 b Std Details
Maximal subgroups
Maximal subgroups of M24
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| M23 | 10 200 960 | 24 | Program: Standard generators Program: Generators |
| M22:2 | 887 040 | 276 | Program: Generators |
| 24:A8 | 322 560 | 759 | Program: Generators |
| M12:2 | 190 080 | 1 288 | Program: Generators |
| 26:3.S6 | 138 240 | 1 771 | Program: Generators |
| L3(4):S3 | 120 960 | 2 024 | Program: Generators |
| 26:(L3(2) × S3) | 64 512 | 3 795 | Program: Generators |
| L2(23) | 6 072 | 40 320 | Program: Generators |
| L2(7) | 168 | 1 457 280 | Program: Generators |
Conjugacy classes
Notes
(M24) Problems of algebraic conjugacy are not yet dealt with.
Conjugacy classes of M24
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 244 823 040 |
Omitted owing to length. | |
| 2A | 21 504 | 4A 4B 6A 8A 12A 14A 14B |
ababababbababababbababababbababababb |
| 2B | 7 680 | 4C 6B 10A 12B |
(ababbababb)3 |
| 3A | 1 080 | 6A 12A 15A 15B |
abababbabababbabababbabababb |
| 3B | 504 | 6B 12B 21A 21B |
ababbababbababbababb |
| 4A | 384 | 12A |
(abababb)3 |
| 4B | 128 | 8A |
ababababbababababb |
| 4C | 96 | 12B |
(ababb)3 |
| 5A | 60 | 10A 15A 15B |
abababbababbabababbababb |
| 6A | 24 | 12A |
abababbabababb |
| 6B | 24 | 12B |
ababbababb |
| 7A | 42 | 7B3 14A 14B 21A 21B |
(ababababbabbababababbabb)3 |
| 7B | 42 | 7A3 14A 14B 21A 21B |
ababababbabbababababbabb |
| 8A | 16 |
ababababb | |
| 10A | 20 |
abababbababb | |
| 11A | 11 |
ababababbabbabb | |
| 12A | 12 |
abababb | |
| 12B | 12 |
ababb | |
| 14A | 14 | 14B3 |
(ababababbabb)3 |
| 14B | 14 | 14A3 |
ababababbabb |
| 15A | 15 | 15B7 |
abababababb |
| 15B | 15 | 15A7 |
(abababababb)7 |
| 21A | 21 | 21B5 |
ababababababb |
| 21B | 21 | 21A5 |
(ababababababb)5 |
| 23A | 23 | 23B5 |
ab |
| 23B | 23 | 23A5 |
(ab)5 |
Download words for class representatives.