Order = 1092 = 22.3.7.13.
Mult = 2.
Out = 2.
Porting notes
Porting incomplete.Standard generators
Standard generators of L2(13) are a, b where a has order 2, b has order 3 and ab has order 13.
Standard generators of 2.L2(13) = SL2(13) are preimages A, B where B has order 3 and AB has order 13.
Standard generators of L2(13).2 are c, d where c has order 2 (necessarily in class 2B), d has order 4 and cd has order 13.
Standard generators of 2.L2(13).2 are preimages C, D where D has order 13.
Automorphisms
| Group | Automorphism | Order in Out(G) | Description |
|---|---|---|---|
| L2(13) | Outer automorphism | 2 | (No program) |
Black box algorithms
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| L2(13) | 〈〈 a, b | some conditions 〉〉 | Download |
Representations
Representations of L2(13)
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- Permutation representations:
Number of points ID Generators Description Link 14 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 C 7 a Std Details 0 C 7 b Std Details 0 C 12 a Std Details 0 C 12 b Std Details 0 C 12 c Std Details 0 Z 13 Std Details 0 Z 14 a Std Details 0 Z 14 b Std Details 0 Z 14 c Std Details 0 Z 36 Std Details Char Ring Dimension ID Generators Description Link 2 GF(4) 6 a Std Details 2 GF(4) 6 b Std Details 2 GF(8) 12 a Std Details 2 GF(8) 12 b Std Details 2 GF(8) 12 c Std Details 2 GF(2) 14 Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 13 Std Details Char Ring Dimension ID Generators Description Link 7 GF(7) 12 Std Details Char Ring Dimension ID Generators Description Link 13 GF(13) 3 Std Details 13 GF(13) 5 Std Details 13 GF(13) 7 Std Details 13 GF(13) 9 Std Details 13 GF(13) 11 Std Details 13 GF(13) 13 Std Details
Representations of 2.L2(13)
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- Matrix representations
Char Ring Dimension ID Generators Description Link 3 GF(3) 6 a Std Details 3 GF(3) 6 b Std Details 3 GF(27) 12 a Std Details 3 GF(27) 12 b Std Details 3 GF(27) 12 c Std Details 3 GF(3) 14 Std Details Char Ring Dimension ID Generators Description Link 7 GF(49) 6 a Std Details 7 GF(49) 6 b Std Details 7 GF(7) 14 a Std Details 7 GF(49) 14 b Std Details 7 GF(49) 14 c Std Details Char Ring Dimension ID Generators Description Link 13 GF(13) 2 Std Details 13 GF(13) 4 Std Details 13 GF(13) 6 Std Details 13 GF(13) 8 Std Details 13 GF(13) 10 Std Details 13 GF(13) 12 Std Details
Representations of L2(13).2
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- Permutation representations:
Number of points ID Generators Description Link 14 Std Details 78 Std Details 91 a Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 12 a Std Details 2 GF(2) 14 Std Details 13 GF(13) 3 a Std Details
Representations of 2.L2(13).2
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- Matrix representations
Char Ring Dimension ID Generators Description Link 3 GF(3) 12 a Std Details
Conjugacy classes
Conjugacy classes of L2(13)
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 1 092 |
(abababbabababb)3 | |
| 2A | 12 |
(abababb)3 | |
| 3A | 6 |
abababbabababb | |
| 6A | 6 |
abababb | |
| 7A | 7 |
ababb | |
| 7B | 7 |
ababbababb | |
| 7C | 7 |
ababbababbababbababb | |
| 13A | 13 |
ab | |
| 13B | 13 |
abab |
Download words for class representatives.
Conjugacy classes of L2(13).2
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 2 184 |
(cdcdcddcdcdcddcdcdcddcdcdcdd)3 | |
| 2A | 24 |
(cdcdcddcdcdcdd)3 | |
| 3A | 12 |
cdcdcddcdcdcddcdcdcddcdcdcdd | |
| 6A | 12 |
cdcdcddcdcdcdd | |
| 7A | 14 |
cddcddcddcddcddcddcddcdd | |
| 7B | 14 |
cddcdd | |
| 7C | 14 |
cddcddcddcdd | |
| 13A | 13 |
cd | |
| 2B | 28 |
(cdd)7 | |
| 4A | 12 |
(cdcdcdd)3 | |
| 12A | 12 |
(cdcdcdd)7 | |
| 12B | 12 |
cdcdcdd | |
| 14A | 14 |
cdd | |
| 14B | 14 |
(cdd)9 | |
| 14C | 14 |
(cdd)3 |
Download words for class representatives.