Order = 443520 = 27.32.5.7.11.
Mult = 12.
Out = 2.
This group has too much information to appear conveniently on one page. We have decided to split it into subpages.
Porting notes
Porting incomplete.Standard generators
Standard generators of M22:2 are c, d where c is in class 2B, d is in class 4C and cd has order 11.
Standard generators of 2.M22.2 are preimages C, D where CD has order 11.
Standard generators of 3.M22:2 are preimages C, D where CD has order 11.
Standard generators of 4.M22.2 are preimages C, D where CD has order 11.
Standard generators of 6.M22.2 are preimages C, D where CD has order 11.
Standard generators of 12.M22.2 are preimages C, D where CD has order 11.
Black box algorithms
Finding generators
| Group | Algorithm | File |
|---|---|---|
| M22:2 | Download |
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| M22:2 | 〈〈 c, d | o(c) = 2, o(d) = 4, o(cd) = 11, o(cdcdd) = 10 〉〉 | Download |
Presentations
| Group | Presentation | Link |
|---|---|---|
| M22:2 | 〈 c, d | c2 = d4 = (cd)11 = (cd2)6 = [c, d]4 = (cdcdcd2cd2)3 = 1 〉 | Details |
Representations
Representations of M22:2
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 22 Std Details 77 Std Details 231 Std Details 330 Std Details 352 Std Details 616 Std Details 672 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 10 a Std Details 2 GF(2) 10 b Std Details 2 GF(2) 34 Std Details 2 GF(2) 98 Std Details 2 GF(2) 140 Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 21 a Std Details 3 GF(9) 45 a Std Details 3 GF(9) 45 b Std Details 3 GF(3) 55 a Std Details 3 GF(3) 98 Std Details 3 GF(3) 99 a Std Details 3 GF(3) 210 a Std Details 3 GF(3) 231 a Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 21 a Std Details 5 GF(25) 45 a Std Details 5 GF(25) 45 b Std Details 5 GF(5) 55 a Std Details 5 GF(5) 98 a Std Details 5 GF(5) 133 a Std Details 5 GF(5) 210 a Std Details 5 GF(5) 385 a Std Details 5 GF(5) 560 Std Details Char Ring Dimension ID Generators Description Link 7 GF(7) 21 a Std Details 7 GF(7) 45 a Std Details 7 GF(7) 54 a Std Details 7 GF(7) 154 a Std Details 7 GF(7) 210 a Std Details 7 GF(7) 231 a Std Details 7 GF(7) 385 a Std Details 7 GF(7) 560 Std Details Char Ring Dimension ID Generators Description Link 11 GF(11) 20 a Std Details 11 GF(11) 45 a Std Details 11 GF(11) 45 b Std Details 11 GF(11) 55 a Std Details 11 GF(11) 99 a Std Details 11 GF(11) 154 a Std Details 11 GF(11) 190 a Std Details 11 GF(11) 231 a Std Details 11 GF(11) 385 a Std Details
Representations of 2.M22.2
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 3 GF(9) 10 b Std Details 5 GF(25) 10 b Std Details 7 GF(7) 10 a Std Details 11 GF(11) 10 b Std Details
Representations of 3.M22:2
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 12 Std Details
Representations of 4.M22.2
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 7 GF(7) 32 Std Details
Representations of 6.M22.2
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 11 GF(11) 72 Std Details
Representations of 12.M22.2
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 11 GF(11) 48 Std Details
Maximal subgroups
Maximal subgroups of M22:2
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| M22 | Program: Generators Program: Generators |
||
| L3(4):22 | Program: Generators Program: Generators |
||
| 24:S6 | Program: Generators |
||
| 25:S5 | Program: Generators |
||
| 23:L3(2) × 2 | Program: Generators Program: Generators |
||
| Aut(A6) = A6.22 | Program: Generators |
||
| L2(11):2 | Program: Generators Program: Generators |
Conjugacy classes
Conjugacy classes of M22:2
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 887 040 |
cddcdcddcddcdcddcddcdcddcddcdcdd |
|
| 2A | 768 | 4A 4B 6A 8A 4C 4D 8B 12A |
cddcdcddcddcdcdd |
| 3A | 72 | 6A 6B 12A |
cddcdd |
| 4A | 64 | 8A 8B |
cddccdcdcdcddcddcddccdcdcdcddcdd |
| 4B | 32 |
cddcdcdd |
|
| 5A | 10 | 10A |
cdcddcdcdd |
| 6A | 24 | 12A |
cdcdcddcdcdcdd |
| 7A | 14 | 7B3 14A 14B |
(ccdcdcdcddcddccdcdcdcddcdd)3 |
| 7B | 14 | 7A3 14A 14B |
ccdcdcdcddcddccdcdcdcddcdd |
| 8A | 16 |
cddccdcdcdcddcdd |
|
| 11A | 11 | cd |
|
| 2B | 2 688 | 6B 14A 14B |
(cdd)3 |
| 2C | 640 | 10A |
(cdcdd)5 |
| 4C | 96 | 12A |
(cdcdcdd)3 |
| 4D | 64 |
dcddccdcdcdcddcdd |
|
| 6B | 12 | cdd |
|
| 8B | 16 |
cdcdcdcdd |
|
| 10A | 10 |
cdcdd |
|
| 12A | 12 |
cdcdcdd |
|
| 14A | 14 | 14B3 |
(ccdcdcdcddcdd)3 |
| 14B | 14 | 14A3 |
ccdcdcdcddcdd |
Download words for class representatives.