Order = 32736 = 25.3.11.31.
Mult = 1.
Out = 5.
Porting notes
Porting incomplete.Standard generators
Standard generators of L2(32) are a, b where a has order 2, b has order 3 and ab is in class 31A/B/C/D/E. Alternatively: a has order 2, b has order 3, ab has order 31 and ababb has order 11.
Standard generators of L2(32):5 are c, d where c has order 2, d has order 5, cd has order 15, cdd has order 15, cdcdd has order 15, cdcdcdd has order 15 and cdcdcdcddcdcddcdd has order 11.
Notes
- (L2(32):5) The conditions ensure that d is conjugate to the field automorphism x -> x16.
Black box algorithms
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| L2(32) | 〈〈 a, b | some conditions 〉〉 | Download |
Presentations
| Group | Presentation | Link |
|---|---|---|
| L2(32) | 〈 a, b | a2 = b3 = (ab)31 = (ab)4(ab−1abab−1)3(ab)4(ab−1)2 = 1 〉 | Details |
| L2(32):5 | 〈 c, d | c2 = d5 = cdcdcd−2cd−1cd−2cdcd−2cd−1 cd2cd−2 = (cd)4cd−2(cd2)3cd−2cd2cd−1cdcd2 = 1 〉 | Details |
Representations
Representations of L2(32)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 33 Std Details 496 Std Details 528 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z 31 a Std Details 0 Z 32 Std Details 0 Z[z31] 33 a Std Details 0 C 33 b Std Details 0 C 33 c Std Details 0 C 33 d Std Details 0 C 33 e Std Details 0 C 33 f Std Details 0 C 33 g Std Details 0 C 33 h Std Details 0 C 33 i Std Details 0 C 33 j Std Details 0 C 33 k Std Details 0 C 33 l Std Details 0 C 33 m Std Details 0 C 33 n Std Details 0 C 33 o Std Details 0 Z 155 Std Details 0 Z 310 Std Details Char Ring Dimension ID Generators Description Link 2 GF(32) 2 a Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 31 a Std Details Char Ring Dimension ID Generators Description Link 11 GF(11) 31 b Std Details Char Ring Dimension ID Generators Description Link 31 GF(31) 32 Std Details
Representations of L2(32):5
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 33 Std Details 496 Std Details 528 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 10 Std Details 2 GF(2) 20 a Std Details 2 GF(2) 20 b Std Details 2 GF(2) 32 Std Details 2 GF(2) 40 a Std Details 2 GF(2) 40 b Std Details 2 GF(2) 80 Std Details
Conjugacy classes
Conjugacy classes of L2(32)
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 32 736 | aa |
|
| 2A | 32 | a |
|
| 3A | 33 |
(ababababb)11 |
|
| 11A | 33 |
(ababababbababababbababababbababababb)3 |
|
| 11B | 33 |
Omitted
owing to
length. |
|
| 11C | 33 |
Omitted
owing to
length. |
|
| 11D | 33 |
(ababababb)3 |
|
| 11E | 33 |
(ababababbababababb)3 |
|
| 31A | 31 | ab |
|
| 31B | 31 |
abab |
|
| 31C | 31 |
abababab |
|
| 31D | 31 |
abababababababab |
|
| 31E | 31 |
abababababababababababababababab |
|
| 31F | 31 |
(abababab)9 |
|
| 31G | 31 |
(abababababababab)9 |
|
| 31H | 31 |
(abababababababababababababababab)9 |
|
| 31I | 31 |
(ab)9 |
|
| 31J | 31 |
(abab)9 |
|
| 31K | 31 |
(abab)3 |
|
| 31L | 31 |
(abababab)3 |
|
| 31M | 31 |
(abababababababab)3 |
|
| 31N | 31 |
(abababababababababababababababab)3 |
|
| 31O | 31 |
(ab)3 |
|
| 33A | 33 |
(ababababb)10 |
|
| 33B | 33 |
(ababababb)13 |
|
| 33C | 33 |
(ababababb)7 |
|
| 33D | 33 |
(ababababb)14 |
|
| 33E | 33 |
(ababababb)5 |
|
| 33F | 33 |
ababababb |
|
| 33G | 33 |
ababababbababababb |
|
| 33H | 33 |
ababababbababababbababababbababababb |
|
| 33I | 33 |
Omitted
owing to
length. |
|
| 33J | 33 |
Omitted
owing to
length. |
Download words for class representatives.
Conjugacy classes of L2(32):5
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 163 680 |
(cdcddcddd)10 |
|
| 2A | 160 |
(cdcddcddd)5 |
|
| 3A | 165 |
(cd)5 |
|
| 11A | 33 |
(cdcdcddd)3 |
|
| 31A | 31 |
(cdcddcdd)3 |
|
| 31F | 31 |
cdcddcdd |
|
| 31K | 31 |
(cdcddcdd)9 |
|
| 33A | 33 |
cdcdcddd |
|
| 33F | 33 |
(cdcdcddd)5 |
|
| 5A | 30 |
(cdcddcddd)16 |
|
| 5A' | 30 |
(cdcddcddd)2 |
|
| 5A'' | 30 |
(cdcddcddd)8 |
|
| 5A''' | 30 |
(cdcddcddd)4 |
|
| 10A | 10 |
cdcddcddd |
|
| 10A' | 10 |
(cdcddcddd)27 |
|
| 10A'' | 10 |
(cdcddcddd)3 |
|
| 10A''' | 10 |
(cdcddcddd)9 |
|
| 15A | 15 | cd |
|
| 15A' | 15 |
(cd)2 |
|
| 15A'' | 15 |
(cd)8 |
|
| 15A''' | 15 |
(cd)4 |
Download words for class representatives.