Order = 168 = 23.3.7.
Mult = 2.
Out = 2.
Porting notes
Porting incomplete.Standard generators
Standard generators of L3(2) = L2(7) are a, b where a has order 2, b has order 3 and ab has order 7.
Standard generators of SL2(7) = 2.L2(7) are preimages A, B where B has order 3 and AB has order 7.
Standard generators of L2(7):2 = PGL2(7) = L3(2):2 are c, d where c is in class 2B, d has order 3, cd has order 8 (necessarily in class 8A) and cdcdd has order 4.
Standard generators of 2.PGL2(7) = 2.L2(7).2 are preimages C, D where D has order 3.
Standard generators of 2.L2(7).2i = 2.L2(7):2 are preimages C, D where D has order 3.
Black box algorithms
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| L3(2) | 〈〈 a, b | some conditions 〉〉 | Download |
Presentations
| Group | Presentation | Link |
|---|---|---|
| L3(2) | 〈 a, b | a2 = b3 = (ab)7 = [a, b]4 = 1 〉 | Details |
| SL2(7) | 〈 A, B | A4 = [A2,B] = B3 = (AB)7 = B2[A,B]4 = 1 〉 | Details |
| L2(7):2 | 〈 c, d | c2 = d3 = (cd)8 = [c, d]4 = 1 〉 | Details |
Representations
Representations of L3(2)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 7 a Std Details 7 b Std Details 8 Std Details 14 a Std Details 14 b Std Details 21 Std Details 24 Std Details 28 Std Details 42 a Std Details 42 b Std Details 42 c Std Details 56 Std Details 84 Std Details 168 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 C 3 a Std Details 0 C 3 b Std Details 0 Z 6 a Std Details 0 Z 6 b Std Details 0 Z 7 Std Details 0 Z 8 Std Details Char Ring Dimension ID Generators Description Link 2 GF(2) 3 a Std Details 2 GF(2) 3 b Std Details 2 GF(2) 8 Std Details Char Ring Dimension ID Generators Description Link 3 GF(9) 3 a Std Details 3 GF(9) 3 b Std Details 3 GF(3) 6 a Std Details 3 GF(3) 6 b Std Details 3 GF(3) 7 Std Details Char Ring Dimension ID Generators Description Link 7 GF(7) 3 Std Details 7 GF(7) 5 Std Details 7 GF(7) 7 Std Details
Representations of SL2(7)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 16 Std Details 48 Std Details 112 Std Details 336 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 3 GF(9) 4 a Std Details 3 GF(9) 4 b Std Details 3 GF(9) 6 a Std Details 3 GF(9) 6 b Std Details 3 GF(3) 8 Std Details 3 GF(3) 12 Std Details 7 GF(7) 2 Std Details 7 GF(7) 4 Std Details 7 GF(7) 6 Std Details
Representations of L2(7):2
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- Permutation representations:
Number of points ID Generators Description Link 8 Std Details 14 Std Details 16 Std Details 21 Std Details 24 Std Details 28 Std Details 28 b Std Details 42 a Std Details 42 b Std Details 42 c Std Details 48 Std Details 56 a Std Details 56 b Std Details 56 c Std Details 84 a Std Details 84 b Std Details 84 c Std Details 112 Std Details 168 a Std Details 168 b Std Details 336 Std Details
Representations of 2.PGL2(7)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 32 Std Details 96 Std Details 224 Std Details 672 Std Details
Representations of 2.L2(7).2i
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 16 a Std Details 16 b Std Details 32 Std Details 48 a Std Details 48 b Std Details 96 Std Details 112 a Std Details 112 b Std Details 224 Std Details 336 Std Details 672 Std Details
Maximal subgroups
Maximal subgroups of L3(2)
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| S4 | Program: Generators Program: Generators Program: Generators | ||
| S4 | Program: Generators Program: Generators Program: Generators | ||
| 7:3 = F21 | Program: Generators Program: Generators Program: Generators |
Maximal subgroups of L2(7):2
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| L2(7) | Program: Standard generators | ||
| 7:6 = F42 | Program: Generators | ||
| D16 | Program: Generators | ||
| D12 | Program: Generators |