Order = 3420 = 22.32.5.19.
Mult = 2.
Out = 2.
Porting notes
Porting incomplete.Standard generators
Standard generators of L2(19) are a, b where a has order 2, b has order 3 and ab has order 19.
Standard generators of 2.L2(19) = SL2(19) are preimages A, B where B has order 3 and AB has order 19.
Standard generators of L2(19):2 = PGL2(19) are c, d where c is in class 2B, d has order 3, cd has order 20 and cdcdd has order 5.
Standard generators of 2.L2(19).2 are preimages C, D where D has order 3.
Black box algorithms
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| L2(19) | 〈〈 a, b | some conditions 〉〉 | Download |
Presentations
| Group | Presentation | Link |
|---|---|---|
| L2(19) | 〈 a, b | a2 = b3 = (ababab−1)5 = [a, bab(ab−1)3abab] = 1 〉 | Details |
| L2(19):2 | 〈 c, d | c2 = d3 = (cd)20 = [c, d]5 = ((cd)4(cd−1)3)2 = 1 〉 | Details |
Representations
Representations of L2(19)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 20 Std Details 57 a Std Details 57 b Std Details 171 Std Details 190 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 C 18 a Std Details 0 C 18 b Std Details 0 C 18 c Std Details 0 C 18 d Std Details 0 Z 18 e Std Details 0 Z 19 Std Details 0 Z 20 a Std Details 0 Z 36 a Std Details Char Ring Dimension ID Generators Description Link 2 GF(4) 9 a Std Details 2 GF(4) 9 b Std Details 2 GF(4) 18 a Std Details 2 GF(4) 18 b Std Details 2 GF(2) 20 a Std Details 2 GF(8) 20 b Std Details 2 GF(8) 20 c Std Details 2 GF(8) 20 d Std Details Char Ring Dimension ID Generators Description Link 3 GF(9) 9 a Std Details 3 GF(9) 9 b Std Details 3 GF(9) 18 a Std Details 3 GF(9) 18 b Std Details 3 GF(9) 18 c Std Details 3 GF(9) 18 d Std Details 3 GF(3) 19 Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 9 a Std Details 5 GF(5) 9 b Std Details 5 GF(5) 18 Std Details 5 GF(5) 20 a Std Details 5 GF(125) 20 b Std Details 5 GF(125) 20 c Std Details 5 GF(125) 20 d Std Details Char Ring Dimension ID Generators Description Link 19 GF(19) 3 Std Details 19 GF(19) 5 Std Details 19 GF(19) 7 Std Details 19 GF(19) 9 Std Details 19 GF(19) 11 Std Details 19 GF(19) 13 Std Details 19 GF(19) 15 Std Details 19 GF(19) 17 Std Details 19 GF(19) 19 Std Details
Representations of 2.L2(19)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 40 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 19 GF(19) 2 Std Details
Representations of L2(19):2
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 20 Std Details 114 Std Details 171 Std Details 190 Std Details 285 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 19 GF(19) 3 Std Details
Representations of 2.L2(19).2i
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 3 GF(3) 20 Details 5 GF(5) 20 Details 19 GF(19) 2 a Details
Conjugacy classes
Conjugacy classes of L2(19)
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 3 420 |
(ababb)9 |
|
| 2A | 20 |
(abababbabb)5 |
|
| 3A | 9 |
(ababb)3 |
|
| 5A | 10 |
abababbabbabababbabbabababbabbabababbabb |
|
| 5B | 10 |
abababbabbabababbabb |
|
| 9A | 9 |
ababb |
|
| 9B | 9 |
ababbababb |
|
| 9C | 9 |
ababbababbababbababb |
|
| 10A | 10 |
abababbabb |
|
| 10B | 10 |
(abababbabb)3 |
|
| 19A | 19 | ab |
|
| 19B | 19 |
abab |
Download words for class representatives.
Conjugacy classes of L2(19):2
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 6 840 |
(cdcdcddcdcddcdcdcddcdcdd)9 |
|
| 2A | 40 |
(cdcd)5 |
|
| 3A | 18 |
(cdcdcddcdcddcdcdcddcdcdd)3 |
|
| 5A | 20 |
cdcdcdcdcdcdcdcd |
|
| 5B | 20 |
cdcdcdcd |
|
| 9A | 18 |
cdcdcddcdcddcdcdcddcdcdd |
|
| 9B | 18 |
cdcdcddcdcddcdcdcddcdcddcdcdcddcdcddcdcdcddcdcdd |
|
| 9C | 18 |
Omitted
owing to
length. |
|
| 10A | 20 |
cdcd |
|
| 10B | 20 |
(cdcd)3 |
|
| 19A | 19 |
cdcdcddcdcdcddcdcdd |
|
| 2B | 36 |
(cdcdcddcdcdd)9 |
|
| 4A | 20 |
(cd)5 |
|
| 6A | 18 |
(cdcdcddcdcdd)3 |
|
| 18A | 18 |
(cdcdcddcdcdd)25 |
|
| 18B | 18 |
(cdcdcddcdcdd)5 |
|
| 18C | 18 |
cdcdcddcdcdd |
|
| 20A | 20 |
(cd)27 |
|
| 20B | 20 | cd |
|
| 20C | 20 |
(cd)3 |
|
| 20D | 20 |
(cd)9 |
Download words for class representatives.