Order = 372000 = 25.3.53.31.
Mult = 1.
Out = 2.
Porting notes
Porting incomplete.Standard generators
Type I generators of L3(5) are a, b where a has order 3, b is in class 5A and has order 20.
Type II generators of L3(5) are x, y where x has order 2, y has order 3, xy has order 31 and xyxyy has order 5.
Standard generators of L3(5):2 are c, d where c is in class 2B, d is in class 4D and cd has order 12.
Black box algorithms
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| L3(5) | 〈〈 a, b | o(a) = 3, o(b) = 5, o(ab) = 20, o(a2babab) = 10 〉〉 | Download |
Presentations
| Group | Presentation | Link |
|---|---|---|
| L3(5) | 〈 a, b | a3 = b5 = aba−1baba−1b2ab−2a−1b2 = abab−2(a−1b2a−1b−2)3 = 1 〉 | Details |
| L3(5) | 〈 x, y | x2 = y3 = (xy)31 = [x, y]5 = ((xy)5(xy−1)4)2 = 1 〉 | Details |
| L3(5):2 | 〈 c, d | c2 = d4 = (cd)12 = (cdcd2cd2)3 = [d2, cdc]3 = [c, dcdcd−1cdcdcd−1cdcd] = 1 〉 | Details |
Representations
Representations of L3(5)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 31 a Type I Details 31 b Type I Details - Matrix representations
Char Ring Dimension ID Generators Description Link 5 GF(5) 3 a Type I Details 5 GF(5) 3 b Type I Details 5 GF(5) 8 Type I Details
Representations of L3(5):2
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 62 Std Details 186 Std Details 775 Std Details 3100 Std Details 3875 Std Details 4000 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 5 GF(5) 6 Std Details 5 GF(5) 8 Std Details
Maximal subgroups
Maximal subgroups of L3(5)
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| 52:GL2(5) | 12 000 | 31 | |
| 52:GL2(5) | 12 000 | 31 | |
| S5 | 120 | 3 100 | |
| 42:S3 | 96 | 3 875 | |
| F93 = 31:3 | 93 | 4 000 |
Maximal subgroups of L3(5):2
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| L3(5) | 372 000 | 2 | Program: Standard
generators |
| 51+2.[25] | 4 000 | 186 | Program: Generators |
| GL2(5).2 | 960 | 775 | Program: Generators |
| S5 × 2 | 240 | 3 100 | Program: Standard
generators |
| 42:D12 | 192 | 3 875 | Program: Generators |
| F186 = 31:6 | 186 | 4 000 | Program: Generators |