Order = 504 = 23.32.7.
Mult = 1.
Out = 3.
Porting notes
Porting incomplete.Standard generators
Standard generators of L2(8) are a, b where a has order 2, b has order 3 and ab has order 7.
Standard generators of L2(8):3 are c, d where c has order 2, d has order 3, cd has order 9 and cdcdcd−1cdcd−1cd−1 has order 7. Alternatively: c has order 2, d has order 3, cd has order 9 and cdcdcd−1cd−1cdcd−1 has order 2.
Black box algorithms
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| L2(8) | 〈〈 a, b | some conditions 〉〉 | Download |
Representations
Representations of L2(8)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 9 Std Details 28 Std Details 36 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z 7 a Std Details 0 Z[y9, 1/3] 7 b Std Details 0 Z[y9, 1/3] 7 c Std Details 0 Z[y9, 1/3] 7 d Std Details 0 Z 8 Std Details 0 Z[y7] 9 a Std Details 0 Z[y7] 9 b Std Details 0 Z[y7] 9 c Std Details 0 Z 27 Std Details Char Ring Dimension ID Generators Description Link 2 GF(8) 2 a Std Details 2 GF(8) 2 b Std Details 2 GF(8) 2 c Std Details 2 GF(8) 4 a Std Details 2 GF(8) 4 b Std Details 2 GF(8) 4 c Std Details 2 GF(2) 6 Std Details 2 GF(2) 8 Std Details 2 GF(2) 12 Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 7 Std Details 3 GF(27) 9 a Std Details 3 GF(27) 9 b Std Details 3 GF(27) 9 c Std Details 3 GF(3) 27 Std Details Char Ring Dimension ID Generators Description Link 7 GF(7) 7 a Std Details 7 GF(7) 8 Std Details 7 GF(7) 21 Std Details
Representations of L2(8):3
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 9 Std Details 28 Std Details 36 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z 7 Std Details 0 Z 8 Std Details 0 Z 21 Std Details 0 Z 27 Std Details Char Ring Dimension ID Generators Description Link 2 GF(2) 6 Std Details 2 GF(2) 8 Std Details 2 GF(2) 12 Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 7 Std Details 3 GF(3) 27 Std Details Char Ring Dimension ID Generators Description Link 7 GF(7) 7 Std Details 7 GF(7) 8 Std Details 7 GF(7) 21 Std Details
Maximal subgroups
Maximal subgroups of L2(8)
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| 23:7 = F56 | |||
| D18 | |||
| D14 |
Maximal subgroups of L2(8):3
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| L2(8) | |||
| 23:7:3 = 23:F21 | |||
| 9:6 | |||
| 7:6 = F42 |
Conjugacy classes
Conjugacy classes of L2(8)
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 504 |
(aaa)2 |
|
| 2A | 8 | aaa |
|
| 3A | 9 |
(ababb)3 |
|
| 7A | 7 | ab |
|
| 7B | 7 |
(ab)2 |
|
| 7C | 7 |
(ab)4 |
|
| 9A | 9 |
ababb |
|
| 9B | 9 |
(ababb)2 |
|
| 9C | 9 |
(ababb)4 |
Download words for class representatives.
Conjugacy classes of L2(8):3
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 1 512 |
(cdcdcdd)6 |
|
| 2A | 24 |
(cdcdcdd)3 |
|
| 3A | 27 |
(cdcdd)3 |
|
| 7A | 7 |
cdcdcddcdd |
|
| 9A | 9 |
cdcdd |
|
| 3B | 18 |
(cdcdcdd)4 |
|
| 3B' | 18 |
(cdcdcdd)2 |
|
| 6A | 6 |
cdcdcdd |
|
| 6A' | 6 |
(cdcdcdd)5 |
|
| 9D | 9 | cd |
|
| 9D' | 9 |
(cd)2 |
Download words for class representatives.