Order = 9828 = 2^{2}.3^{3}.7.13.

Mult = 2.

Out = 6.

## Porting notes

Porting incomplete.## Standard generators

Standard generators of L_{2}(27) are *a*, *b* where *a* has order 2, *b* has order 3 and *a**b* has order 7.

Standard generators of 2.L_{2}(27) = SL_{2}(27) are preimages *A*, *B* where *B* has order 3 and *A**B* has order 7.

Standard generators of L_{2}(27).2 are *c*, *d* where *c* has order 2 (necessarily in class 2B), *d* has order 4 and *c**d* has order 7.

Standard generators of 2L_{2}(27).2 are preimages *C*, *D* where *C**D* has order 7.

Standard generators of L_{2}(27).3 are *e*, *f* where *e* has order 2, *f* has order 3, *e**f* has order 9, *e**f**e**f**f* has order 7 and *e**f**e**f**e**f**e**f**f**e**f**e**f**f**e**f**f* has order 9.

Standard generators of 2L_{2}(27).3 are preimages *E*, *F* where *F* has order 3 and *E**F* has order 9.

Standard generators of L_{2}(27).6 are *g*, *h* where *g* has order 2 (necessarily in class 2B), *h* has order 3 (necessarily in class 3C), *g**h* has order 24 and *g**h**g**h**h* has order 13.

Standard generators of 2L_{2}(27).6 are preimages *G*, *H* where *H* has order 3.

## Black box algorithms

### Checking generators (semi-presentations)

Group | Semi-presentation | File |
---|---|---|

L_{2}(27)
| 〈〈 a, b | some conditions 〉〉
| Download |

## Representations

### Representations of L_{2}(27)

- View detailed report.
- 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.L_{2}(27)

- View detailed report.
- 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 L_{2}(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.