Order = 1876896 =
25.32.73.19.
Mult = 3.
Out = S3.
Porting notes
Porting incomplete.Standard generators
Standard generators of L3(7) are a, b where a has order 2, b has order 3, ab has order 19 and ababb has order 6.
Standard generators of 3.L3(7) are preimages A, B where A has order 2 and AB has order 19.
Standard generators of L3(7).2 are c, d where c is in class 2B, d is in class 4B, cd has order 19 and cdcdd has order 8.
Standard generators of 3.L3(7).2 are preimages C, D where CD has order 19.
Black box algorithms
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| L3(7) | 〈〈 a, b | some conditions 〉〉 | Download |
Representations
Representations of L3(7)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 57 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z 56 Std Details 0 Z 57 Std Details 0 Z 152 a Std Details 0 Z 152 b Std Details 0 Z 152 c Std Details Char Ring Dimension ID Generators Description Link 2 GF(2) 152 a Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 55 Std Details 3 GF(3) 57 Std Details 3 GF(3) 96 a Std Details 3 GF(3) 96 b Std Details 3 GF(3) 96 c Std Details 3 GF(3) 342 a Std Details 3 GF(3) 399 Std Details Char Ring Dimension ID Generators Description Link 7 GF(7) 8 Std Details 7 GF(7) 10 a Std Details 7 GF(7) 10 b Std Details 7 GF(7) 27 Std Details
Representations of 3.L3(7)
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 7 GF(7) 3 Std Details
Representations of L3(7).2
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 56 Std Details 2 GF(2) 152 Std Details 2 GF(2) 342 Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 55 Std Details 3 GF(3) 57 Std Details 3 GF(3) 96 Std Details 3 GF(3) 192 Std Details 3 GF(9) 342 b Std Details 3 GF(9) 342 c Std Details 3 GF(3) 399 Std Details Char Ring Dimension ID Generators Description Link 7 GF(7) 8 Std Details 7 GF(7) 20 Std Details 7 GF(7) 27 Std Details 7 GF(7) 37 Std Details 7 GF(7) 56 Std Details 7 GF(7) 70 Std Details 7 GF(7) 308 Std Details 7 GF(7) 343 Std Details
Representations of 3.L3(7).2
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 7 GF(7) 6 Std Details
Maximal subgroups
Maximal subgroups of L3(7)
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| 72:2.L2(7):2 | |||
| 72:2.L2(7):2 | |||
| L2(7).2 | |||
| L2(7).2 | |||
| L2(7).2 | |||
| (3 x A4)2 | |||
| 32:Q8 | |||
| 19:3 |
Maximal subgroups of L3(7).2
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| L3(7) | Program: Generators |
||
| 71+2:(3 x D8) | Program: Generators |
||
| 2.(2 x L2(7)).2 | |||
| L2(7)×2 | |||
| S3 x S4 | |||
| 32:SD16 | |||
| 19:6 |
Conjugacy classes
Conjugacy classes of L3(7)
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 1 876 896 |
Omitted
owing to
length. |
|
| 2A | 672 |
abababbabababbabababbabababbabababbabababbabababbabababb |
|
| 3A | 36 |
ababbababb |
|
| 4A | 16 |
abababbabababbabababbabababb |
|
| 6A | 12 |
ababb |
|
| 7A | 686 |
abbabababbababbabbabababbababb |
|
| 7B | 49 |
aabababbabbabababbababb |
|
| 7C | 49 |
abbabababbababbabababbababbabbabababbababb |
|
| 7D | 49 |
baabababbabbabababbababbaabababbabbabababbababb |
|
| 8A | 16 |
(abababbabababb)3 |
|
| 8B | 16 |
abababbabababb |
|
| 14A | 14 |
abbabababbababb |
|
| 16A | 16 |
(abababb)27 |
|
| 16B | 16 |
(abababb)3 |
|
| 16C | 16 |
abababb |
|
| 16D | 16 |
(abababb)9 |
|
| 19A | 19 |
abababab |
|
| 19B | 19 |
Omitted
owing to
length. |
|
| 19C | 19 | ab |
|
| 19D | 19 |
abababababababab |
|
| 19E | 19 |
abababababababababababababababab |
|
| 19F | 19 |
abab |
Download words for class representatives.
Conjugacy classes of L3(7).2
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 3 753 792 |
(cdcdd)8 |
|
| 2A | 1 344 |
(cdcdd)4 |
|
| 3A | 72 |
(cccdcddcdcdcddcdcdcdd)2 |
|
| 4A | 32 |
(cdcdd)2 |
|
| 6A | 24 |
(cccdcddcdcdcddcdd)2 |
|
| 7A | 1 372 |
(cdcdcdd)4 |
|
| 7B | 98 |
(ccdcddcdcdcdd)2 |
|
| 7C | 49 |
ccdcddcdcdcddcdcdcddcccdcddcdcdcdd |
|
| 8A | 32 |
(cdd)2 |
|
| 8B | 32 |
(ccdcddcdcdcddcdd)2 |
|
| 14A | 28 |
(cdcdcdd)2 |
|
| 16A | 16 |
(ccdcddcdcdcddcdd)3 |
|
| 16C | 16 |
ccdcddcdcdcddcdd |
|
| 19A | 19 | cd |
|
| 19C | 19 |
(cd)2 |
|
| 19E | 19 |
(cd)4 |
|
| 2B | 672 |
(cccdcddcdcdcddcdcdcdd)3 |
|
| 4B | 672 |
(cccdcddcdcdcddcdd)3 |
|
| 6D | 12 |
cccdcddcdcdcddcdcdcdd |
|
| 8C | 16 |
cdcdd |
|
| 12B | 12 |
cccdcddcdcdcddcdd |
|
| 14B | 14 |
ccdcddcdcdcdd |
|
| 16E | 16 | cdd |
|
| 16F | 16 |
(cdd)3 |
|
| 28A | 28 |
cdcdcdd |
|
| 28B | 28 |
(cdcdcdd)5 |
Download words for class representatives.