Order = 126000 = 24.32.53.7.
Mult = 3.
Out = S3.
Porting notes
Fully copied from version 2. 30/7/06.Standard generators
Standard generators of U3(5) are a, b where a has order 3, b is in class 5A and ab has order 7.
Standard generators of 3.U3(5) are preimages A, B where B has order 5 and AB has order 7.
Standard generators of U3(5):2 are c, d where c is in class 2B, d is in class 4A, cd has order 10 and cdcdddcdd has order 2.
Standard generators of 3.U3(5):2 are preimages C, D where D has order 4.
Standard generators of U3(5):3 are e, f where e is in class 2A, f is in class 3C/C' and ef has order 15.
Standard generators of 3.U3(5).3 are preimages E, F where E has order 2.
Standard generators of U3(5):S3 are g, h where g is in class 2B, h is in class 3C, gh has order 8 and ghghh has order 12.
Standard generators of 3.U3(5).S3 are preimages G, H.
Presentations
| Group | Presentation | Link |
|---|---|---|
| U3(5) | 〈 a, b | a3 = b5 = (ab)7 = (ab−1)7 = aba−1b2aba−1bab2a−1b = 1 〉 | Details |
| U3(5):2 | 〈 c, d | c2 = d4 = (cd)10 = (cdcd−1cd2)2 = [c, dcd]4 = (cdcdcdcd2)7 = 1 〉 | Details |
| U3(5):3 | 〈 e, f | e2 = f3 = (ef)15 = [e, f]6 = [e, (ef)4(ef−1)3(ef)4] = [e, (ef)3(ef−1)6(ef)3] = 1 〉 | Details |
| U3(5):S3 | 〈 g, h | g2 = h3 = (gh)8 = [g, h]12 = (ghghgh−1)6 = [g, hgh]6 = (ghghgh−1ghgh−1)10 = 1 〉 | Details |
Representations
Representations of U3(5)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 50 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 C 20 Std Details 0 Z 21 Std Details 0 Z 28 a Std Details 0 Z 28 b Std Details 0 Z 28 c Std Details 0 Z 84 Std Details 0 Z 105 Std Details 0 Z 125 Std Details 0 Z 126 a Std Details 0 Z 252 Std Details 0 Z 288 Std Details
Representations of 3.U3(5)
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 0 Q(ω) 21 a Std Details
Representations of U3(5):2
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 50 Std Details 126 Std Details 175 Std Details 525 Std Details 750 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 20 Std Details 2 GF(2) 28 Std Details 2 GF(2) 56 Std Details 2 GF(2) 104 Std Details 2 GF(2) 288 Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 20 a Std Details 3 GF(3) 21 a Std Details 3 GF(3) 28 a Std Details 3 GF(3) 56 Std Details 3 GF(3) 84 a Std Details 3 GF(3) 126 a Std Details 3 GF(3) 252 Std Details 3 GF(3) 288 Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 8 a Std Details 5 GF(5) 19 a Std Details 5 GF(5) 20 Std Details 5 GF(5) 63 a Std Details 5 GF(5) 70 Std Details 5 GF(5) 125 a Std Details Char Ring Dimension ID Generators Description Link 7 GF(7) 20 a Std Details 7 GF(7) 21 a Std Details 7 GF(7) 28 a Std Details 7 GF(7) 56 Std Details 7 GF(7) 84 a Std Details 7 GF(7) 105 a Std Details 7 GF(7) 124 a Std Details 7 GF(7) 126 a Std Details 7 GF(7) 252 Std Details
Maximal subgroups
Maximal subgroups of U3(5)
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| A7 | |||
| A7 | |||
| A7 | |||
| 51+2:8 | |||
| M10 = A6.2 | |||
| M10 = A6.2 | |||
| M10 = A6.2 | |||
| 2.S5 = 2.A5.2 |
Maximal subgroups of U3(5):2
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| U3(5) | Program: Standard generators | ||
| S7 | Program: Generators Program: Standard generators | ||
| 51+2:8:2 | Program: Generators | ||
| A6.22 | Program: Standard generators | ||
| 2.S5.2 | Program: Generators | ||
| L2(7):2 | Program: Standard generators |
Maximal subgroups of U3(5):3
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| U3(5) | |||
| 51+2:24 | |||
| 2.S5 × 3 | |||
| 62:S3 | |||
| 32:2A4 | |||
| 7:3 × 3 |
Maximal subgroups of U3(5):S3
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| U3(5):3 | |||
| U3(5):2 | |||
| 51+2:24:2 | |||
| (3 × 2.S5).2 | |||
| 62:D12 | |||
| 32:2S4 | |||
| (7:3 × 3):2 |