Order = 9828 = 22.33.7.13.
Mult = 2.
Out = 6.
Porting notes
Porting incomplete.Standard generators
Standard generators of L2(27) are a, b where a has order 2, b has order 3 and ab has order 7.
Standard generators of 2.L2(27) = SL2(27) are preimages A, B where B has order 3 and AB has order 7.
Standard generators of L2(27).2 are c, d where c has order 2 (necessarily in class 2B), d has order 4 and cd has order 7.
Standard generators of 2L2(27).2 are preimages C, D where CD has order 7.
Standard generators of L2(27).3 are e, f where e has order 2, f has order 3, ef has order 9, efeff has order 7 and efefefeffefeffeff has order 9.
Standard generators of 2L2(27).3 are preimages E, F where F has order 3 and EF has order 9.
Standard generators of L2(27).6 are g, h where g has order 2 (necessarily in class 2B), h has order 3 (necessarily in class 3C), gh has order 24 and ghghh has order 13.
Standard generators of 2L2(27).6 are preimages G, H where H has order 3.
Black box algorithms
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| L2(27) | 〈〈 a, b | some conditions 〉〉 | Download |
Representations
Representations of L2(27)
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- Permutation representations:
Number of points ID Generators Description Link 28 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z[ω] 13 a Std Details 0 Z[ω] 13 b Std Details 0 Z 26 g Std Details 0 Z 27 Std Details 0 C 28 a Std Details 0 C 28 b Std Details 0 C 28 c Std Details 0 C 28 d Std Details 0 C 28 e Std Details 0 C 28 f Std Details 0 Z 78 a Std Details Char Ring Dimension ID Generators Description Link 2 GF(4) 13 a Std Details 2 GF(4) 13 b Std Details Char Ring Dimension ID Generators Description Link 3 GF(27) 3 a Std Details 3 GF(3) 27 Std Details Char Ring Dimension ID Generators Description Link 7 GF(7) 26 Std Details Char Ring Dimension ID Generators Description Link 13 GF(13) 27 Std Details
Representations of 2.L2(27)
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- Matrix representations
Char Ring Dimension ID Generators Description Link 3 GF(27) 2 a Std Details 3 GF(27) 2 b Std Details 3 GF(27) 2 c Std Details
Conjugacy classes
Conjugacy classes of L2(27)
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 9 828 |
(b)3 | |
| 2A | 28 |
(abababbabb)7 | |
| 3A | 27 |
b | |
| 3B | 27 |
bb | |
| 7A | 14 |
abababbabbabababbabb | |
| 7B | 14 |
Omitted owing to length. | |
| 7C | 14 |
abababbabbabababbabbabababbabbabababbabb | |
| 13A | 13 |
ababbababbababbababbababbababbababbababb | |
| 13B | 13 |
ababbababb | |
| 13C | 13 |
Omitted owing to length. | |
| 13D | 13 |
ababb | |
| 13E | 13 |
Omitted owing to length. | |
| 13F | 13 |
ababbababbababbababb | |
| 14A | 14 |
(abababbabb)9 | |
| 14B | 14 |
abababbabb | |
| 14C | 14 |
(abababbabb)3 |
Download words for class representatives.