Order = 6072 = 23.3.11.23.
Mult = 2.
Out = 2.
Porting notes
Porting incomplete.Standard generators
Standard generators of L2(23) are a, b where a has order 2, b has order 3 and ab has order 23.
Standard generators of 2.L2(23) = SL2(23) are preimages A, B where B has order 3 and AB has order 23.
Standard generators of L2(23):2 are c, d where c has order 2, d has order 3, cd has order 22 and cdcdd has order 4.
Standard generators of 2.L2(23).2 are C, D where D has order 3.
Standard generators of 2.L2(23).2 (isoclinic) are preimages C, D where D has order 3.
Black box algorithms
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| L2(23) | 〈〈 a, b | some conditions 〉〉 | Download |
Representations
Representations of L2(23)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 24 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 C 11 a Std Details 0 C 11 b Std Details 0 Z 22 a Std Details 0 Z 22 b Std Details 0 Z 22 c Std Details 0 C 22 d Std Details 0 C 22 e Std Details 0 Z 23 Std Details 0 C 24 a Std Details 0 C 24 b Std Details 0 C 24 c Std Details 0 C 24 d Std Details 0 C 24 e Std Details Char Ring Dimension ID Generators Description Link 2 GF(2) 11 a Std Details 2 GF(2) 11 b Std Details 2 GF(2) 22 Std Details 2 GF(32) 24 a Std Details 2 GF(32) 24 b Std Details 2 GF(32) 24 c Std Details 2 GF(32) 24 d Std Details 2 GF(32) 24 e Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 22 a Std Details Char Ring Dimension ID Generators Description Link 11 GF(11) 23 Std Details Char Ring Dimension ID Generators Description Link 23 GF(23) 3 Std Details 23 GF(23) 5 Std Details 23 GF(23) 7 Std Details 23 GF(23) 9 Std Details 23 GF(23) 11 Std Details 23 GF(23) 13 Std Details 23 GF(23) 15 Std Details 23 GF(23) 17 Std Details 23 GF(23) 19 Std Details 23 GF(23) 21 Std Details 23 GF(23) 23 Std Details
Representations of 2.L2(23)
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- Matrix representations
Char Ring Dimension ID Generators Description Link 23 GF(23) 2 Std Details
Representations of L2(23):2
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- Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 22 a Std Details 23 GF(23) 3 a Std Details
Representations of 2.L2(23).2
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 23 GF(23) 2 Std Details
Representations of 2.L2(23).2 (isoclinic)
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- Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 23 Std Details
Maximal subgroups
Maximal subgroups of L2(23)
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| 23:11 | |||
| S4 | |||
| S4 | |||
| D24 | |||
| D22 |
Maximal subgroups of L2(23):2
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| L2(23) | |||
| 23:22 | |||
| D48 | |||
| D44 |
Conjugacy classes
Conjugacy classes of L2(23)
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 6 072 |
(ababbababbababbababb)3 |
|
| 2A | 24 |
(ababbababb)3 |
|
| 3A | 12 |
ababbababbababbababb |
|
| 4A | 12 |
(ababb)3 |
|
| 6A | 12 |
ababbababb |
|
| 11A | 11 |
abababbabababb |
|
| 11B | 11 |
Omitted
owing to
length. |
|
| 11C | 11 |
abababbabababbabababbabababb |
|
| 11D | 11 |
abababb |
|
| 11E | 11 |
abababbabababbabababbabababbabababbabababbabababbabababb |
|
| 12A | 12 |
(ababb)5 |
|
| 12B | 12 |
ababb |
|
| 23A | 23 | ab |
|
| 23B | 23 |
(ab)11 |
Download words for class representatives.
Conjugacy classes of L2(23):2
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 12 144 |
(cdcdcddcdcdcddcdcdcddcdcdcddcdcdcddcdcdcddcdcdcddcdcdcdd)3 |
|
| 2A | 48 |
(cdcdcddcdcdcddcdcdcddcdcdcdd)3 |
|
| 3A | 24 |
cdcdcddcdcdcddcdcdcddcdcdcddcdcdcddcdcdcddcdcdcddcdcdcdd |
|
| 4A | 24 |
(cdcdcddcdcdcdd)3 |
|
| 6A | 24 |
cdcdcddcdcdcddcdcdcddcdcdcdd |
|
| 11A | 22 |
cdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcd |
|
| 11B | 22 |
cdcdcdcd |
|
| 11C | 22 |
Omitted
owing to
length. |
|
| 11D | 22 |
cdcdcdcdcdcdcdcd |
|
| 11E | 22 |
cdcd |
|
| 12A | 24 |
cdcdcddcdcdcdd |
|
| 12B | 24 |
(cdcdcddcdcdcdd)5 |
|
| 23A | 23 |
ccdcdcdcddcdcddcddcdcdcddcd |
|
| 2B | 44 |
(cd)11 |
|
| 8A | 24 |
(cdcdcdd)3 |
|
| 8B | 24 |
(cdcdcdd)9 |
|
| 22A | 22 |
(cd)27 |
|
| 22B | 22 |
(cd)81 |
|
| 22C | 22 | cd |
|
| 22D | 22 |
(cd)3 |
|
| 22E | 22 |
(cd)9 |
|
| 24A | 24 |
(cdcdcdd)77 |
|
| 24B | 24 |
(cdcdcdd)11 |
|
| 24C | 24 |
(cdcdcdd)55 |
|
| 24D | 24 |
cdcdcdd |
Download words for class representatives.