Order = 5515776 = 29.34.7.19.
Mult = 3.
Out = 3 × S3.
Porting notes
Porting incomplete.Standard generators
Standard generators of U3(8) are a, b where a has order 2, b has order 3 (necessarily in class 3C) and ab has order 19.
Standard generators of 3.U3(8) = SU3(8) are preimages A, B where A has order 2 and AB has order 19.
Standard generators of U3(8):2 are c, d where c is in class 2B, d is in class 3C, cd has order 8 and cdcdcddcdcddcdd has order 9.
Standard generators of 3.U3(8):2 are preimages C, D where CDCDD has order 19.
Standard generators of U3(8):31 are e, f where e has order 2, f is in class 3D/E/F/D'/E'/F', ef has order 12, efeff has order 7 and efefeffefeffeff has order 7.
Standard generators of 3.U3(8):31 are preimages E, F where E has order 2 and F has order 3.
Standard generators of U3(8):6 are g, h where g is in class 2B, h is in class 3D/D'/EF/EF' (i.e. an outer element of order 3), gh has order 18, ghghh has order 19 and ghghghhghghhghh has order 9.
Standard generators of U3(8):32 are i, j where i has order 2, j is in class 3G/G' and ij has order 9.
Standard generators of U3(8).33 are k, l where k has order 2, l is in class 9K/L/M/K'/L'/M', kl has order 9, kl2 has order 9, kl3 has order 6, kl4 has order 18 and klkl2kl4 has order 9.
Standard generators of U3(8):S3 are m, n where m is in class 2B, n is in class 3G/G', mn has order 8, mnmnn has order 9 and (mn)3mn2mnmn2mn2 has order 14.
Standard generators of U3(8).32 are o, p where o is in class 3DEF/DEF', p is in class 9EFG/EFG', op has order 9, op2 has order 9, op3 has order 12, op4 has order 9 and opo2p2 has order 7.
Standard generators of U3(8).(S3 × 3) are q, r where q is in class 2B, r is in class 9KLM/KLM', qr has order 6, qrqrr has order 3 and qrqr2qr4 has order 6.
Presentations
| Group | Presentation | Link |
|---|---|---|
| U3(8) | 〈 a, b | a2 = b3 = (ab)19 = [a,b]9 = [a,bab]3 = (abababab−1)3ab−1ab(ab−1)3ab(ab−1)2 = (((ab)4(ab−1)3)2ab−1)2 = 1 〉 | Details |
Representations
Representations of U3(8)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Q(ω) 57 a Std Details 0 Q(ω) 57 b Std Details 0 Z 114 Std Details 0 Z 133 a Std Details 0 Z 133 b Std Details 0 Z 133 c Std Details 3 GF(3) 56 Std Details
Representations of 3.U3(8)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 4617 Std Details 32832 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(64) 3 a Std Details
Representations of U3(8):2
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details
Representations of U3(8):31
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details
Representations of U3(8):6
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 24 Std Details 2 GF(2) 54 a Std Details 2 GF(2) 54 b Std Details 2 GF(2) 192 Std Details 2 GF(2) 432 Std Details 2 GF(2) 512 Std Details 3 GF(3) 56 a Std Details 3 GF(3) 133 a Std Details 3 GF(3) 266 Std Details
Representations of U3(8):32
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details
Representations of U3(8).33
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details
Representations of U3(8):S3
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details
Representations of U3(8).32
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details
Representations of U3(8).(S3 × 3)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 513 Std Details 3648 Std Details