Order = 60 = 22.3.5.
Mult = 2.
Out = 2.
Porting notes
Porting incomplete.Standard generators
Standard generators of A5 = L2(5) = L2(4) are a, b where a has order 2, b has order 3 and ab has order 5.
Standard generators of 2.A5 = SL2(5) are preimages A, B where B has order 3 and AB has order 5.
Standard generators of S5 = A5:2 are c, d where c is in class 2B, d has order 4 and cd has order 5.
Standard generators of 2.S5 are preimages C, D where CD has order 5.
Standard generators of 2.S5i are preimages C, D where CD has order 5.
Black box algorithms
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| A5 | 〈〈 a, b | some conditions 〉〉 | Download |
Presentations
| Group | Presentation | Link |
|---|---|---|
| A5 | 〈 a, b | a2 = b3 = (ab)5 = 1 〉 | Details |
| 2.A5 | 〈 A, B | A4 = [A2, B] = B3 = (AB)5 = 1 〉 | Details |
| S5 | 〈 c, d | c2 = d4 = (cd)5 = [c, d]3 = 1 〉 | Details |
| 2.S5 | 〈 C, D | C2 = D8 = [C, D4] = (CD)5 = [C, D]3 = 1 〉 | Details |
| 2.S5i | 〈 C, D | C4 = C−2D4 = (CD)5 = [C, D]3 = 1 〉 | Details |
Representations
Representations of A5
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 5 Std Natural representation Details 6 Std Details 10 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z[b5] 3 a Std Details 0 Z[b5] 3 b Std Details 0 Z 4 Std Details 0 Z 5 Std Details 0 Z 6 Std Details Char Ring Dimension ID Generators Description Link 2 GF(4) 2 a Std Natural representation as L2(4) Details 2 GF(4) 2 b Std Field automorph of the above Details 2 GF(2) 4 a Std Natural representation as O4−(2) Details 2 GF(2) 4 b 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) 4 Std Details 3 GF(3) 6 Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 3 Std Details 5 GF(5) 5 Std Details
Representations of 2.A5
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 24 Std Details 40 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 H 1 a Std Over quaternions over Q(b5) Details 0 Z[z5] 2 a Std Details 0 Z[i] 4 a Std Details 0 Z[i6] 4 a Std Details 0 Z[i] 6 Std Details 0 Z[ω] 6 Std Details 0 Z 8 b Std Details 0 Z 12 Std Details Char Ring Dimension ID Generators Description Link 2 GF(4) 5 a Std Details 2 GF(4) 5 b Std Details 2 GF(2) 8 Std Details 2 GF(2) 9 Std Details 2 GF(2) 10 Std Details Char Ring Dimension ID Generators Description Link 3 GF(9) 2 a Std Details 3 GF(9) 2 b Std Details 3 GF(3) 4 Std Details 3 GF(3) 6 Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 2 Std Details 5 GF(5) 4 Std Details Char Ring Dimension ID Generators Description Link 7 GF(49) 2 a Std Details 7 GF(49) 2 b Std Details 7 GF(7) 4 a Std Details 7 GF(7) 4 b Std Details 7 GF(7) 6 Std Details
Representations of S5
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 5 Std Details 6 Std Details 10 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 4 a Std Details 2 GF(2) 4 b Std Details 3 GF(3) 4 Std Details 3 GF(3) 6 Std Details 5 GF(5) 3 Std Details 5 GF(5) 5 Std Details
Representations of 2.S5
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 40 a Std Details 40 b Std Details 48 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z[ω] 4 b Std Details 0 Z 8 a Std Details 3 GF(3) 4 Std Details 3 GF(3) 6 Std Details 5 GF(25) 2 Std Details 5 GF(25) 4 a Std Details 5 GF(5) 4 b Std Details 5 GF(5) 8 Std Details
Representations of 2.S5i
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 48 Std Details 80 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z[ω] 4 a Std Details 0 Z[i2] 4 a Std Basis 1. Details 0 Z[i5] 4 a Std Basis 2. Details 3 GF(3) 4 Std Details 3 GF(9) 6 Std Details 3 GF(3) 12 Std Details 5 GF(25) 2 Std Details 5 GF(25) 4 a Std Details 5 GF(5) 4 b Std Details 5 GF(5) 8 Std Details
Maximal subgroups
Maximal subgroups of A5
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| A4 | 12 | 5 | Program: Generators |
| D10 | 10 | 6 | Program: Generators |
| S3 | 6 | 10 | Program: Generators |
Maximal subgroups of S5
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| A5 | 60 | 2 | Program: Standard
generators |
| S4 | 24 | 5 | Program: Generators |
| 5:4 | 20 | 6 | Program: Generators |
| S3 × 2 = D12 | 12 | 10 | Program: Generators |