Order = 660 = 22.3.5.11.
Mult = 2.
Out = 2.
Porting notes
Porting incomplete.Standard generators
Standard generators of L2(11) are a, b where a has order 2, b has order 3 and ab has order 11.
Standard generators of 2.L2(11) = SL2(11) are preimages A, B where B has order 3 and AB has order 11.
Standard generators of L2(11):2 = PGL2(11) are c, d where c is in class 2B, d has order 3, cd has order 10 (necessarily in class 10A) and cdcdd has order 11.
Standard generators of 2.L2(11).2 = 2.PGL2(11) are preimages C, D where D has order 3.
Black box algorithms
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| L2(11) | 〈〈 a, b | o(a) = 2, o(b) = 3, o(ab) = 11 〉〉 | Download |
| L2(11):2 | 〈〈 c, d | o(c) = 2, o(d) = 3, o(cd) = 10, o(cdcd2) = 11 〉〉 | Download |
Presentations
| Group | Presentation | Link |
|---|---|---|
| L2(11) | 〈 a, b | a2 = b3 = (ab)11 = [a, babab]2 = 1 〉 | Details |
| L2(11):2 | 〈 c, d | c2 = d3 = (cd)10 = (cdcdcd−1)4 = [c, dcdcd−1cd−1cd−1cdcd] = 1 〉 | Details |
Representations
Representations of L2(11)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 11 a Std Details 11 b Std Details 12 Std Details 55 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 C 5 a Std Details 0 C 5 b Std Details 0 Z 10 a Std Details 0 Z 10 b Std Details 0 Z 10 c Std Details 0 Z 11 Std Details 0 C 12 a Std Details 0 C 12 b Std Details Char Ring Dimension ID Generators Description Link 2 GF(4) 5 a Std Details 2 GF(4) 5 b Std Details 2 GF(2) 10 Std Details 2 GF(2) 10 b Std Details 2 GF(4) 12 a Std Details 2 GF(4) 12 b Std Details 2 GF(2) 24 Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 5 a Std Details 3 GF(3) 5 b Std Details 3 GF(3) 10 Std Details 3 GF(9) 12 a Std Details 3 GF(9) 12 b Std Details 3 GF(3) 24 Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 5 a Std Details 5 GF(5) 5 b Std Details 5 GF(5) 10 a Std Details 5 GF(5) 10 b Std Details 5 GF(5) 11 Std Details Char Ring Dimension ID Generators Description Link 11 GF(11) 3 Std Details 11 GF(11) 5 Std Details 11 GF(11) 7 Std Details 11 GF(11) 9 Std Details 11 GF(11) 11 Std Details
Representations of 2.L2(11)
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 11 GF(11) 2 Std Details 11 GF(11) 4 Std Details 11 GF(11) 6 Std Details 11 GF(11) 8 Std Details 11 GF(11) 10 Std Details
Representations of L2(11):2
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 12 Std Details 22 Std Details 55 a Std Details 55 b Std Details 66 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 10 a Std Details 2 GF(2) 10 b Std Details 2 GF(4) 12 a Std Details 2 GF(4) 12 b Std Details 11 GF(11) 3 Std Details 11 GF(11) 5 Std Details 11 GF(11) 7 Std Details 11 GF(11) 9 Std Details 11 GF(11) 11 Std Details
Maximal subgroups
Maximal subgroups of L2(11)
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| A5 | Program: Generators |
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| A5 | Program: Generators |
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| F55 = 11:5 | Program: Generators |
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| D12 | Program: Generators |
Maximal subgroups of L2(11):2
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| L2(11) | Program: Generators |
||
| F110 = 11:10 | Program: Generators |
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| S4 | Program: Generators |
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| D24 | Program: Generators |
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| D20 | Program: Generators |
Conjugacy classes
Conjugacy classes of L2(11)
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 660 |
(ababababbababababb)3 |
|
| 2A | 12 |
(ababababb)3 |
|
| 3A | 6 |
ababababbababababb |
|
| 5A | 5 |
abababb |
|
| 5B | 5 |
abababbabababb |
|
| 6A | 6 |
ababababb |
|
| 11A | 11 | ab |
|
| 11B | 11 |
abab |
Download words for class representatives.
Conjugacy classes of L2(11):2
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 1 320 |
(cdcdcddcdcdd)12 |
|
| 2A | 24 |
(cdcdcddcdcdd)6 |
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| 3A | 12 |
(cdcdcddcdcdd)4 |
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| 5A | 10 |
(cd)4 |
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| 5B | 10 |
(cd)2 |
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| 6A | 12 |
(cdcdcddcdcdd)2 |
|
| 11A | 11 |
cdcdd |
|
| 2B | 20 |
(cd)5 |
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| 4A | 12 |
(cdcdcddcdcdd)3 |
|
| 10A | 10 | cd |
|
| 10B | 10 |
(cd)3 |
|
| 12A | 12 |
cdcdcddcdcdd |
|
| 12B | 12 |
(cdcdcddcdcdd)5 |
Download words for class representatives.