Order = 25920 = 26.34.5.
Mult = 2.
Out = 2.
Porting notes
Porting incomplete.Standard generators
Standard generators of U4(2) = S4(3) are a, b where a is in class 2A, b has order 5 and ab has order 9.
Standard generators of 2.U4(2) = Sp4(3) are preimages A, B where B has order 5 and AB has order 9.
Standard generators of U4(2):2 = S4(3):2 are c, d where c is in class 2C, d has order 9 and cd has order 10.
Standard generators of 2.U4(2):2 are preimages C, D where D has order 9.
Black box algorithms
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| U4(2) | 〈〈 a, b | some conditions 〉〉 | Download |
Presentations
| Group | Presentation | Link |
|---|---|---|
| U4(2) | 〈 a, b | a2 = b5 = (ab)9 = [a, b]3 = [a, bab]2 = 1 〉 | Details |
Representations
Representations of U4(2)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 27 Std Details 36 Std Details 40 a Std Details 40 b Std Details 45 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z[ω] 5 a Std Basis 0. Details 0 Z[ω] 5 a Std Basis 1: Written over Z[ω]. Details 0 Z[ω] 5 a Std Basis 2: Written over Z[ω]. Details 0 Q(ω) 5 a Std Basis 3: Unitary. Details 0 Z[ω] 5 b Std Details 0 Z 6 Std Details 0 Z[ω] 10 Std Details 0 Z[ω] 10 a Std Details 0 Z[ω] 10 b Std Details 0 Z 15 a Std Details 0 Z 15 b Std Details 0 Z 20 Std Details 0 Z 24 Std Details 0 Z 30 a Std Details 0 Z[ω] 30 b Std Details 0 Z[ω] 30 c Std Details 0 Z[ω] 40 a Std Details 0 Z[ω] 40 b Std Details 0 Z[ω] 45 a Std Details 0 Z[ω] 45 b Std Details 0 Z 60 Std Details 0 Z 64 Std Details 0 Z 80 Std Details 0 Z 81 Std Details Char Ring Dimension ID Generators Description Link 2 GF(4) 4 a Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 5 Std Details 3 GF(3) 10 Std Details 3 GF(3) 14 Std Details 3 GF(3) 25 Std Details 3 GF(3) 81 Std Details Char Ring Dimension ID Generators Description Link 5 GF(25) 5 a Std Details 5 GF(5) 6 Std Details 5 GF(25) 10 b Std Details
Representations of 2.U4(2)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 80 Std Details 240 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z[ω] 4 a Std Details 0 Z[ω] 4 b Std Details 0 Z 8 Std Details 0 Z[ω] 20 a Std Details 0 Z[ω] 20 b Std Details 0 Z[ω] 20 c Std Details 0 Z[ω] 36 a Std Details 0 Z[ω] 36 b Std Details 0 Z[ω] 60 b Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 4 Std Details 3 GF(3) 16 Std Details 3 GF(3) 40 Std Details
Representations of U4(2):2
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 27 Std Details 36 Std Details 40 a Std Details 40 b Std Details 45 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z 6 a Std Basis 0. Details 0 Z 6 a Std Basis 1. Details 0 Z 81 a Std Details 2 GF(2) 6 Std Details 2 GF(2) 8 Std Details 2 GF(2) 14 Std Details 2 GF(2) 40 Std Details 2 GF(2) 64 Std Details 3 GF(3) 5 Std Details
Representations of 2.U4(2):2
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 240 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 3 GF(3) 4 Std Details
Conjugacy classes
Conjugacy classes of U4(2)
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 25 920 |
abaababbababbabaababbababbabaababbababbabaababbababb | |
| 2A | 576 |
(ababb)3 | |
| 2B | 96 |
abaababbababbabaababbababb | |
| 3A | 648 |
abbabbabbabbabbabbabbabb | |
| 3B | 648 |
abbabbabbabb | |
| 3C | 108 |
ababbababb | |
| 3D | 54 |
bababaababbababbbababaababbababb | |
| 4A | 48 |
(abb)3 | |
| 4B | 8 |
abaababbababb | |
| 5A | 5 |
b | |
| 6A | 72 |
abbabb | |
| 6B | 72 |
(abbabb)5 | |
| 6C | 36 |
(ababb)5 | |
| 6D | 36 |
ababb | |
| 6E | 18 |
bababaababbababb | |
| 6F | 12 |
aababbababb | |
| 9A | 9 |
ab | |
| 9B | 9 |
abab | |
| 12A | 12 |
(abb)5 | |
| 12B | 12 |
abb |
Download words for class representatives.
Conjugacy classes of U4(2):2
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 51 840 |
(cdcdcddcdcdddcdd)4 | |
| 2A | 1 152 |
(cdd)4 | |
| 2B | 192 |
(cdcdcddcdcdddcdd)2 | |
| 3A | 648 |
(cdcdd)4 | |
| 3C | 216 |
(ccdcdddcddd)2 | |
| 3D | 108 |
(cddcdcdddcdd)2 | |
| 4A | 96 |
(cdd)2 | |
| 4B | 16 |
cdcdcddcdcdddcdd | |
| 5A | 10 |
(cd)2 | |
| 6A | 72 |
(cdcdd)2 | |
| 6C | 36 |
ccdcdddcddd | |
| 6E | 36 |
cddcdcdddcdd | |
| 6F | 24 |
(cdcdcdd)2 | |
| 9A | 9 |
d | |
| 12A | 12 |
cdcdd | |
| 2C | 1 440 |
(ccdcdcddcdcdddcddcddcdcdddcdd)3 | |
| 2D | 96 |
(cdcdddcdd)3 | |
| 4C | 96 |
(cdcdcdd)3 | |
| 4D | 32 |
dcdcdcdd | |
| 6G | 36 |
ccdcdcddcdcdddcddcddcdcdddcdd | |
| 6H | 36 |
dcdd | |
| 6I | 12 |
cdcdddcdd | |
| 8A | 8 |
cdd | |
| 10A | 10 |
cd | |
| 12C | 12 |
cdcdcdd |
Download words for class representatives.