Order = 14880 = 25.3.5.31.
Mult = 2.
Out = 2.
Porting notes
Porting incomplete.Standard generators
Standard generators of L2(31) are a, b where a has order 2, b has order 3 and ab has order 31.
Standard generators of 2.L2(31) are preimages A, B where B has order 3 and AB has order 31.
Standard generators of L2(31):2 = PGL2(31) are c, d where c has order 2, d has order 3, cd has order 10 and cdcdd has order 8.
Standard generators of 2.L2(31).2 are C, D where D has order 3.
Black box algorithms
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| L2(31) | 〈〈 a, b | some conditions 〉〉 | Download |
Presentations
| Group | Presentation | Link |
|---|---|---|
| L2(31) | 〈 a, b | a2 = b3 = [a,b]3ab[a,b][a,bab][a,b−1abab] = 1 〉 | Details |
| L2(31):2 | 〈 c, d | c2 = d3 = (cd)10 = [c, d]8 = (cd(cdcdcd−1)3)2 = 1 〉 | Details |
Representations
Representations of L2(31)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 32 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 C 15 a Std Details 0 C 15 b Std Details 0 Z 30 a Std Details 0 C 30 b Std Details 0 C 30 c Std Details 0 Z 30 h Std Details 0 Z 31 Std Details 0 Z 32 a Std Details 0 C 32 b Std Details 0 C 32 c Std Details 0 C 32 d Std Details 0 C 32 e Std Details 0 C 32 f Std Details 0 C 32 g Std Details 0 Z 60 Std Details 0 Z 120 Std Details Char Ring Dimension ID Generators Description Link 2 GF(2) 15 a Std Details 2 GF(2) 15 b Std Details 2 GF(2) 32 Std Details 2 GF(4) 32 b Std Details 2 GF(4) 32 c Std Details 2 GF(16) 32 d Std Details 2 GF(16) 32 e Std Details 2 GF(16) 32 f Std Details 2 GF(16) 32 g Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 31 Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 31 Std Details Char Ring Dimension ID Generators Description Link 31 GF(31) 3 Std Details 31 GF(31) 31 Std Details
Representations of 2.L2(31)
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z[z30] 32 a Std Details 5 GF(5) 16 a Std Details 31 GF(31) 2 Std Details
Representations of L2(31):2
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 30 Std Details
Conjugacy classes
Conjugacy classes of L2(31)
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 14 880 |
Omitted owing to length. | |
| 2A | 32 |
abababbabababbabababbabababbabababbabababbabababbabababb | |
| 3A | 15 |
(ababb)5 | |
| 4A | 16 |
abababbabababbabababbabababb | |
| 5A | 15 |
(ababb)3 | |
| 5B | 15 |
(ababbababb)3 | |
| 8A | 16 |
(abababbabababb)3 | |
| 8B | 16 |
abababbabababb | |
| 15A | 15 |
ababbababbababbababbababbababbababbababb | |
| 15B | 15 |
ababb | |
| 15C | 15 |
ababbababb | |
| 15D | 15 |
ababbababbababbababb | |
| 16A | 16 |
(abababb)27 | |
| 16B | 16 |
abababb | |
| 16C | 16 |
(abababb)3 | |
| 16D | 16 |
(abababb)9 | |
| 31A | 31 |
ab | |
| 31B | 31 |
(ab)3 |
Download words for class representatives.