Order = 95040 = 26.33.5.11.
Mult = 2.
Out = 2.
Porting notes
Porting incomplete.Standard generators
Standard generators of M12 are a, b where a is in class 2B, b is in class 3B and ab has order 11.
Standard generators of 2.M12 are preimages A, B where A is in class +2B and B has order 6. Alternatively: A is in class +2B and AB has order 11 or B has order 6 and AB has order 11.
Standard generators of M12:2 are c, d where c is in class 2C, d is in class 3A and cd is in class 12A. Alternatively: c is in class 2C, d is in class 3A, cd has order 12 and cdcdd has order 11.
Standard generators of 2.M12.2 are preimages C, D where D has order 3.
Black box algorithms
Finding generators
Group | Algorithm | File |
---|---|---|
M12 |
| Download |
M12:2 |
| Download |
Checking generators (semi-presentations)
Group | Semi-presentation | File |
---|---|---|
M12 | 〈〈 a, b | o(a) = 2, o(b) = 3, o(ab) = 11, o(ababab2) = 6 〉〉 | Download |
M12:2 | 〈〈 c, d | o(c) = 2, o(d) = 3, o(cd) = 12, o(cdcdcdcd2) = 6 〉〉 | Download |
Presentations
Group | Presentation | Link |
---|---|---|
M12 | 〈 a, b | a2 = b3 = (ab)11 = [a, b]6 = (ababab−1)6 = 1 〉 | Details |
2.M12 | 〈 A, B | A2 = B6 = [B3, A] = (AB)11 = [A, B]6 = (ABABAB−1)6B−3 = [A, BAB]5 = 1 〉 | Details |
M12:2 | 〈 c, d | c2 = d3 = (cd)12 = (cd)5[c, d](cd−1)3cd[c, d−1]2cdcd(cd−1)3[c, d−1] = 1 〉 | Details |
M12:2 | 〈 c, d | c2 = d3 = (cd)12 = (cdcd[c,d−1]4)2 = ((cd)4cd−1)4 = [c, dcd]5 = 1 〉 | Details |
Representations
Representations of M12
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 12 a Std Details 12 b Std Details 66 a Std Details 66 b Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z 11 a Std Details 0 Z 11 b Std Details 0 C 16 a Std Details 0 C 16 b Std Details 0 Z 32 Std Details 0 Z 45 Std Details 0 Z 54 Std Details 0 Z 55 a Std Details 0 Z 55 b Std Details 0 Z 55 c Std Details 0 Z 66 Std Details 0 Z 99 Std Details 0 Z 120 Std Details 0 Z 144 Std Details 0 Z 176 Std Details Char Ring Dimension ID Generators Description Link 2 GF(2) 10 Std Details 2 GF(4) 16 a Std Details 2 GF(4) 16 b Std Details 2 GF(2) 44 Std Details 2 GF(2) 144 Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 10 a Std Details 3 GF(3) 15 b Std Details 3 GF(3) 34 Std Details 3 GF(3) 45 a Std Details 3 GF(3) 45 b Std Details 3 GF(3) 54 Std Details 3 GF(3) 99 Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 11 a Std Details 5 GF(5) 16 b Std Details 5 GF(5) 45 Std Details 5 GF(5) 55 a Std Details 5 GF(5) 55 c Std Details 5 GF(5) 66 Std Details 5 GF(5) 78 Std Details 5 GF(5) 98 Std Details 5 GF(5) 120 Std Details Char Ring Dimension ID Generators Description Link 11 GF(11) 11 a Std Details 11 GF(11) 16 Std Details 11 GF(11) 29 Std Details 11 GF(11) 53 Std Details 11 GF(11) 55 a Std Details 11 GF(11) 55 c Std Details 11 GF(11) 66 Std Details 11 GF(11) 91 Std Details 11 GF(11) 99 Std Details 11 GF(11) 176 Std Details
Representations of 2.M12
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 24 a Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z 12 Std Details 0 Z 120 Std Details 0 Z 220 Std Details 3 GF(3) 6 b Std Details 5 GF(5) 12 Std Details
Representations of M12:2
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 24 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 10 Std Details 2 GF(2) 32 Std Details 2 GF(2) 44 Std Details 2 GF(2) 144 Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 20 a Std Details 3 GF(3) 30 a Std Details 3 GF(3) 34 a Std Details 3 GF(3) 45 a Std Details 3 GF(9) 54 a Std Details 3 GF(3) 90 a Std Details 3 GF(3) 99 a Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 22 a Std Details 5 GF(5) 32 a Std Details 5 GF(5) 45 a Std Details 5 GF(5) 55 a Std Details 5 GF(5) 66 a Std Details 5 GF(5) 78 a Std Details 5 GF(5) 98 a Std Details 5 GF(5) 110 a Std Details 5 GF(25) 120 a Std Details Char Ring Dimension ID Generators Description Link 11 GF(11) 16 a Std Details 11 GF(11) 22 a Std Details 11 GF(11) 29 a Std Details 11 GF(11) 53 a Std Details 11 GF(11) 55 a Std Details 11 GF(11) 66 a Std Details 11 GF(11) 91 a Std Details 11 GF(11) 99 a Std Details 11 GF(11) 110 a Std Details 11 GF(11) 176 a Std Details
Representations of 2.M12.2
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 48 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 3 GF(3) 10 a Std Details 3 GF(3) 10 b Std Details 3 GF(3) 10 c Std Details 3 GF(3) 12 Std Details 3 GF(3) 12 a Std Details 3 GF(3) 88 a Std Details 3 GF(3) 168 a Std Details Char Ring Dimension ID Generators Description Link 5 GF(25) 10 a Std Details 5 GF(25) 10 b Std Details 5 GF(25) 12 a Std Details 5 GF(25) 32 b Std Details 5 GF(5) 120 b Std Details 5 GF(5) 220 a Std Details 5 GF(5) 320 a Std Details Char Ring Dimension ID Generators Description Link 11 GF(11) 10 a Std Details 11 GF(11) 10 b Std Details 11 GF(11) 12 a Std Details 11 GF(11) 32 a Std Details 11 GF(11) 88 a Std Details 11 GF(11) 108 a Std Details 11 GF(11) 220 a Std Details
Maximal subgroups
Maximal subgroups of M12
Subgroup | Order | Index | Programs/reps |
---|---|---|---|
M11 | 7 920 | 12 | Program: Standard generators |
M11 | 7 920 | 12 | Program: Standard generators |
A6.22 | 1 440 | 66 | Program: Generators |
A6.22 | 1 440 | 66 | Program: Generators |
L2(11) | 660 | 144 | Program: Standard generators |
32:2S4 | 432 | 220 | Program: Generators |
32:2S4 | 432 | 220 | Program: Generators |
2 × S5 | 240 | 396 | Program: Generators mapping onto standard generators |
21+4:S3 | 192 | 495 | Program: Generators Program: Generators |
42:D12 | 192 | 495 | Program: Generators |
A4 × S3 | 72 | 1 320 | Program: Generators |
Maximal subgroups of M12:2
Subgroup | Order | Index | Programs/reps |
---|---|---|---|
M12 | Program: Standard generators | ||
L2(11):2 | Program: Generators Program: Generators | ||
L2(11):2 | Program: Generators Program: Generators | ||
(2 × 2 × A5).2. | Program: Generators | ||
21+4:S3.2. | Program: Generators | ||
42:D12.2. | Program: Generators | ||
31+2:D8 | Program: Generators | ||
S4 × S3 | Program: Generators | ||
S5 | Program: Generators |
Conjugacy classes
Conjugacy classes of M12
Conjugacy class | Centraliser order | Power up | Class rep(s) |
---|---|---|---|
1A | 95 040 |
Omitted owing to length. | |
2A | 240 | 6A 10A |
(abababb)3 |
2B | 192 | 4A 4B 6B 8A 8B |
Omitted owing to length. |
3A | 54 | 6B |
ababbababb |
3B | 36 | 6A |
abababbabababb |
4A | 32 | 8A |
ababababbababbabbababababbababbabb |
4B | 32 | 8B |
ababababbabbababbababababbabbababb |
5A | 10 | 10A |
ababababbabbababababbabb |
6A | 12 |
abababb | |
6B | 6 |
ababb | |
8A | 8 |
ababababbababbabb | |
8B | 8 |
ababababbabbababb | |
10A | 10 |
ababababbabb | |
11A | 11 | 11B2 |
ab |
11B | 11 | 11A2 |
abab |
Download words for class representatives.
Conjugacy classes of M12:2
Conjugacy class | Centraliser order | Power up | Class rep(s) |
---|---|---|---|
1A | 190 080 | ||
2A | 480 | 6A 10A 4C 12A | |
2B | 384 | 4A 6B 8A 4B 12B 12C | |
3A | 108 | 6B 12B 12C | |
3B | 72 | 6A 6C 12A | |
4A | 32 | 8A | |
5A | 20 | 10A 10B 10C | |
6A | 24 | 12A | |
6B | 12 | 12B 12C | |
8A | 8 |
cdcdcddcdcdcdcdcddcdcdd | |
10A | 20 |
cdcdcdcddcdcdd | |
11A | 11 |
cdcdd | |
2C | 240 | 6C 10B 10C | |
4B | 48 | 12B 12C | |
4C | 24 | 12A | |
6C | 12 |
cdcdcddcdcddcdcdd | |
10B | 20 | 10C3 | |
10C | 20 | 10B3 |
cdcdcdd |
12A | 12 |
cd | |
12B | 12 | 12C5 | |
12C | 12 | 12B5 |
cdcdcdcddcdcddcdd |
Download words for class representatives.