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

### 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

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 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

## 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

### 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