Order = 4080 = 24.3.5.17.
Mult = 1.
Out = 4.
Porting notes
Porting incomplete.Standard generators
Standard generators of L2(16) are a, b where a has order 2, b has order 3 and ab has order 15.
Standard generators of L2(16):2 are c, d where c is in class 2B, d has order 4 and cd has order 15.
Standard generators of L2(16):4 are e, f where e is in class 2A, f is in class 4B/B', ef has order 8, eff has order 6 and efeffefff has order 4.
Black box algorithms
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| L2(16) | 〈〈 a, b | some conditions 〉〉 | Download |
Presentations
| Group | Presentation | Link |
|---|---|---|
| L2(16) | 〈 a, b | a2 = b3 = (ab)15 = ((ab)5(ab−1)3)2 = 1 〉 | Details |
| L2(16):2 | 〈 c, d | c2 = d4 = [c, dcd]2 = cd(cdcd2)2(cd2cd)2cdcd−1 = 1 〉 | Details |
| L2(16):4 | 〈 e, f | e2 = f4 = (ef)8 = (ef2)6 = (ef)3(ef−1)3(ef)2(ef−1)2(ef)2ef2 = 1 〉 | Details |
Representations
Representations of L2(16)
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- Permutation representations:
Number of points ID Generators Description Link 17 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z 16 Std Details 0 Z 17 a Std Details 0 C 17 b Std Details 0 C 17 c Std Details 0 C 17 d Std Details 0 C 17 e Std Details 0 C 17 f Std Details 0 C 17 g Std Details 0 Z 34 Std Details 0 Z 68 Std Details 0 Z 120 Std Details Char Ring Dimension ID Generators Description Link 2 GF(16) 2 a Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 16 Std Details 3 GF(9) 17 a Std Details 3 GF(9) 17 b Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 16 Std Details Char Ring Dimension ID Generators Description Link 17 GF(17) 15 Std Details 17 GF(17) 17 Std Details 17 GF(17) 34 Std Details 17 GF(17) 68 Std Details
Representations of L2(16):2
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- Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(4) 4 a Std Details
Representations of L2(16):4
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- Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 8 a Std Details 2 GF(2) 8 b Std Details 2 GF(2) 16 a Std Details 2 GF(2) 16 b Std Details 2 GF(2) 32 Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 16 a Std Details 3 GF(9) 60 a Std Details 3 GF(9) 60 b Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 16 a Std Details 5 GF(5) 17 a Std Details 5 GF(25) 60 a Std Details 5 GF(25) 60 b Std Details Char Ring Dimension ID Generators Description Link 17 GF(17) 15 a Std Details 17 GF(17) 17 a Std Details 17 GF(17) 34 a Std Details 17 GF(17) 68 Std Details
Conjugacy classes
Conjugacy classes of L2(16)
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 4 080 |
aa | |
| 2A | 16 |
a | |
| 3A | 15 |
(ab)5 | |
| 5A | 15 |
(abab)3 | |
| 5B | 15 |
(ab)3 | |
| 15A | 15 |
ab | |
| 15B | 15 |
abababab | |
| 15C | 15 |
abab | |
| 15D | 15 |
abababababababab | |
| 17A | 17 |
ababb | |
| 17B | 17 |
ababbababbababbababb | |
| 17C | 17 |
ababbababb | |
| 17D | 17 |
ababbababbababbababbababbababbababbababb | |
| 17E | 17 |
(ababbababb)3 | |
| 17F | 17 |
(ababbababbababbababbababbababbababbababb)3 | |
| 17G | 17 |
(ababbababbababbababb)3 | |
| 17H | 17 |
(ababb)3 |
Download words for class representatives.
Conjugacy classes of L2(16):2
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 8 160 |
cdcdcddcdcdcddcdcdcddcdcdcdd | |
| 2A | 32 |
cdcdcddcdcdcdd | |
| 3A | 30 |
cdcddcdcdd | |
| 5A | 30 |
cddcddcddcdd | |
| 5B | 30 |
cddcdd | |
| 15A | 15 |
cd | |
| 15C | 15 |
cdcd | |
| 17A | 17 |
(cdcdcdddcdcdcddd)3 | |
| 17C | 17 |
(cdcdcddd)3 | |
| 17E | 17 |
cdcdcdddcdcdcddd | |
| 17G | 17 |
cdcdcddd | |
| 2B | 120 |
(cdcdd)3 | |
| 4A | 8 |
cdcdcdd | |
| 6A | 6 |
cdcdd | |
| 10A | 10 |
cdd | |
| 10B | 10 |
(cdd)3 |
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Conjugacy classes of L2(16):4
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 16 320 |
(ef)8 | |
| 2A | 64 |
(ef)4 | |
| 3A | 60 |
(efeff)4 | |
| 5A | 30 |
(efefeffeff)2 | |
| 15A | 15 |
efefeff | |
| 17A | 17 |
(efefff)3 | |
| 17E | 17 |
efefff | |
| 2B | 240 |
(efeff)6 | |
| 4A | 16 |
(ef)2 | |
| 6A | 12 |
(efeff)2 | |
| 10A | 10 |
efefeffeff | |
| 4B | 24 |
(efeff)3 | |
| 4B' | 24 |
(efeff)9 | |
| 8A | 8 |
ef | |
| 8A' | 8 |
(ef)3 | |
| 12A | 12 |
(efeff)7 | |
| 12A' | 12 |
efeff |
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