Order = 50232960 = 27.35.5.17.19.
Mult = 3.
Out = 2.
Porting notes
Porting incomplete.Standard generators
Standard generators of J3 are a, b where a has order 2, b is in class 3A, ab has order 19 and ababb has order 9.
Standard generators of 3.J3 are preimages A, B where A has order 2 and B is in class +3A. Alternatively: A has order 2 and ABABABB has order 17.
Standard generators of J3:2 are c, d where c is in class 2B, d is in class 3A, cd has order 24 and cdcdd has order 9.
Standard generators of 3.J3:2 are preimages C, D where D is in class +3A.
Black box algorithms
Finding generators
Group | Algorithm | File |
---|---|---|
J3 |
| Download |
J3:2 |
| Download |
Checking generators (semi-presentations)
Group | Semi-presentation | File |
---|---|---|
J3 | 〈〈 a, b | o(a) = 2, o(b) = 3, o(ab) = 19, o(ababab2) = 17 〉〉 | Download |
J3:2 | 〈〈 c, d | o(c) = 2, o(d) = 3, o(cd) = 24, o(cdcdcdcd2) = 9 〉〉 | Download |
Presentations
Group | Presentation | Link |
---|---|---|
J3 | 〈 a, b | a2 = b3 = (ab)19 = [a, b]9 = ((ab)6(ab−1)5)2 = ((ababab−1)2abab−1ab−1abab−1)2 = abab(abab−1)3abab(abab−1)4ab−1(abab−1)3 = (ababababab−1abab−1)4 = 1 〉 | Details |
3.J3 | 〈 A, B | A2 = B3 = [A,B]9 = ((AB)6(AB−1)5)2 = ((ABABAB−1)2ABAB−1AB−1ABAB−1)2 = ABAB(ABAB−1)3ABAB(ABAB−1)4AB−1(ABAB−1)3 = ((AB)3(ABAB−1)2)4(AB)19 = 1 〉 | Details |
3.J3 | 〈 A, B | A2 = B3 = [A,B]9 = ((AB)6(AB−1)5)2 = ((ABABAB−1)2ABAB−1AB−1ABAB−1)2 = ABAB(ABAB−1)3ABAB(ABAB−1)4AB−1(ABAB−1)3 = (AB)4(AB−1)2AB(ABABAB−1)2(ABAB−1AB)2AB(AB−1)4(AB)4(AB−1)3 = ((AB)5AB−1(AB)2(AB−1)5AB(AB−1)2)2 = ((AB)5(AB−1ABAB−1)2)3 = 1 〉 | Details |
J3:2 | 〈 c, d | c2 = d3 = (cd)24 = [c, d]9 = (cd(cdcd−1)2)4 = (cdcdcd−1(cdcdcd−1cd−1)2)2 = [c, (dc)4(d−1c)2d]2 = [c, d(cd−1)2(cd)4]2 = 1 〉 | Details |
Representations
Representations of J3
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 6156 Std Details 14688 a Std Details 14688 b Std Details 17442 Std Details 20520 Std Details 23256 Std Details 25840 Std Details 26163 Std Details 43605 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z 170 Std Details Char Ring Dimension ID Generators Description Link 2 GF(4) 78 b Std Details 2 GF(2) 80 Std Details 2 GF(4) 84 a Std Details 2 GF(2) 244 Std Details 2 GF(2) 248 Std Details 2 GF(4) 322 a Std Details 2 GF(2) 966 Std Details Char Ring Dimension ID Generators Description Link 3 GF(9) 18 b Std Details 3 GF(9) 84 b Std Details 3 GF(9) 153 b Std Details 3 GF(3) 324 Std Details 3 GF(3) 934 Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 85 a Std Details 5 GF(5) 323 Std Details 5 GF(5) 646 Std Details 5 GF(5) 816 Std Details Char Ring Dimension ID Generators Description Link 17 GF(17) 85 b Std Details 17 GF(17) 324 Std Details 17 GF(17) 379 Std Details 17 GF(17) 646 c Std Details 17 GF(17) 761 Std Details 17 GF(17) 816 Std Details 17 GF(17) 836 Std Details 17 GF(17) 1292 Std Details Char Ring Dimension ID Generators Description Link 19 GF(19) 85 Std Details 19 GF(19) 110 Std Details 19 GF(19) 214 Std Details 19 GF(19) 323 a Std Details 19 GF(19) 646 a Std Details 19 GF(19) 706 Std Details 19 GF(19) 919 Std Details 19 GF(19) 1001 Std Details
Representations of 3.J3
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 18468 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(4) 9 a Std Details 2 GF(4) 18 a Std Details 2 GF(4) 18 b Std Details 2 GF(4) 126 a Std Details 2 GF(4) 153 a Std Details 2 GF(4) 153 b Std Details 2 GF(4) 324 a Std Details 2 GF(4) 720 a Std Details 2 GF(4) 1008 Std Details Char Ring Dimension ID Generators Description Link 5 GF(25) 18 Std Details 5 GF(5) 36 Std Details 5 GF(25) 153 Std Details 5 GF(25) 171 a Std Details Char Ring Dimension ID Generators Description Link 17 GF(17) 36 a Std Details 17 GF(17) 36 b Std Details 17 GF(17) 342 a Std Details 17 GF(17) 648 a Std Details Char Ring Dimension ID Generators Description Link 19 GF(19) 18 a Std Details 19 GF(19) 18 b Std Details
Representations of J3:2
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 6156 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 80 a Std Details 2 GF(2) 156 a Std Details 2 GF(2) 168 a Std Details 2 GF(2) 244 a Std Details 2 GF(2) 644 a Std Details 2 GF(2) 966 a Std Details Char Ring Dimension ID Generators Description Link 3 GF(3) 36 Std Details 3 GF(3) 36 a Std Details 3 GF(3) 168 a Std Details 3 GF(3) 306 a Std Details 3 GF(3) 324 a Std Details 3 GF(3) 934 a Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 170 a Std Details 5 GF(5) 323 b Std Details 5 GF(5) 646 a Std Details 5 GF(5) 816 a Std Details Char Ring Dimension ID Generators Description Link 17 GF(17) 170 a Std Details 17 GF(17) 324 a Std Details 17 GF(17) 379 a Std Details 17 GF(17) 646 a Std Details 17 GF(17) 761 a Std Details 17 GF(17) 836 a Std Details Char Ring Dimension ID Generators Description Link 19 GF(19) 85 a Std Details 19 GF(19) 110 a Std Details 19 GF(19) 214 a Std Details 19 GF(19) 214 b Std Details 19 GF(19) 646 a Std Details 19 GF(19) 706 a Std Details 19 GF(19) 919 a Std Details 19 GF(19) 1001 a Std Details 19 GF(19) 1214 a Std Details
Representations of 3.J3:2
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 18 Std Details
Maximal subgroups
Maximal subgroups of J3
Subgroup | Order | Index | Programs/reps |
---|---|---|---|
L2(16):2 | Program: Generators | ||
L2(19) | Program: Generators | ||
L2(19) | Program: Generators | ||
24:(3 × A5) | Program: Generators | ||
L2(17) | Program: Generators | ||
(3 × A6):22 | Program: Generators | ||
32+1+2:8 | Program: Generators | ||
21+4:A5 | Program: Generators | ||
22+4:(3 × S3) | Program: Generators |
Maximal subgroups of J3:2
Subgroup | Order | Index | Programs/reps |
---|---|---|---|
J3 | Program: Generators Program: Generators | ||
L2(16):4 | Program: Generators Program: Generators | ||
24:(3 × A5).2 | Program: Generators Program: Generators | ||
L2(17) × 2 | Program: Generators Program: Generators Program: Generators | ||
(3 × M10):2 | Program: Generators Program: Generators | ||
32+1+2:8.2 | Program: Generators Program: Generators | ||
21+4:S5 | Program: Generators Program: Generators | ||
22+4:(S3 × S3) | Program: Generators Program: Generators | ||
19:18 = F342 | Program: Generators Program: Generators |
Conjugacy classes
Conjugacy classes of J3
Conjugacy class | Centraliser order | Power up | Class rep(s) |
---|---|---|---|
1A | 50 232 960 |
Omitted owing to length. | |
2A | 1 920 | 4A 6A 8A 10A 10B 12A |
ababbabababbababbabababbababbabababbababbabababb |
3A | 1 080 | 6A 12A 15A 15B |
Omitted owing to length. |
3B | 243 | 9A 9B 9C |
(ababb)3 |
4A | 96 | 8A 12A |
ababbabababbababbabababb |
5A | 30 | 5B2 10A 10B 15A 15B |
Omitted owing to length. |
5B | 30 | 5A2 10A 10B 15A 15B |
abbababbabababbabbabbababbabababbabb |
6A | 24 | 12A |
abbababbabababbabbabababbabbababbabababbabbabababb |
8A | 8 |
ababbabababb | |
9A | 27 | 9B4 9C2 |
ababbababbababbababb |
9B | 27 | 9A2 9C4 |
ababb |
9C | 27 | 9A4 9B2 |
ababbababb |
10A | 10 | 10B3 |
abbababbabababbabb |
10B | 10 | 10A3 |
(abbababbabababbabb)3 |
12A | 12 |
abbababbabababbabbabababb | |
15A | 15 | 15B2 |
abbababbabababbabbababbabababb |
15B | 15 | 15A2 |
abbababbabababb |
17A | 17 | 17B3 |
(abababb)3 |
17B | 17 | 17A3 |
abababb |
19A | 19 | 19B2 |
ab |
19B | 19 | 19A2 |
abab |
Download words for class representatives.
Conjugacy classes of J3:2
Conjugacy class | Centraliser order | Power up | Class rep(s) |
---|---|---|---|
1A | 100 465 920 | ||
2A | 3 840 | 4A 6A 8A 10A 12A 4B 8B 8C 12B 24A 24B | |
3A | 2 160 | 6A 12A 15A 12B 24A 24B | |
3B | 486 | 9A 9B 9C 6B 18A 18B 18C | |
4A | 192 | 8A 12A 8B 8C 24A 24B | |
5A | 30 | 10A 15A | |
6A | 48 | 12A 12B 24A 24B | |
8A | 16 |
cdcddcdcdcdcdcddcdcdcdcddcdcdcdd | |
9A | 54 | 9B4 9C2 18A 18B 18C | |
9B | 54 | 9A2 9C4 18A 18B 18C | |
9C | 54 | 9A4 9B2 18A 18B 18C | |
10A | 10 |
cdcdcdcddcdcdd | |
12A | 24 | 24A 24B | |
15A | 15 |
cdcdcdcdcddcdcdcdcddcdcdcdd | |
17A | 34 | 17B3 34A 34B | |
17B | 34 | 17A3 34A 34B | |
19A | 19 |
cdcdcddcdd | |
2B | 4 896 | 6B 18A 18B 18C 34A 34B | |
4B | 96 | 12B | |
6B | 18 | 18A 18B 18C | |
8B | 96 | 24A 24B | |
8C | 32 |
cdcddcddcdcdcdcdcddcdcdcdcddcdcdcdd | |
12B | 12 |
cdcdcdcdcddcdcdcdcdd | |
18A | 18 | 18B5 18C7 | |
18B | 18 | 18A7 18C5 |
cdcdcdcdcdd |
18C | 18 | 18A5 18B7 | |
24A | 24 | 24B7 |
cd |
24B | 24 | 24A7 | |
34A | 34 | 34B3 | |
34B | 34 | 34A3 |
cdcdcdd |
Download words for class representatives.