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

Representations of 3.J3

Representations of J3:2

Representations of 3.J3:2

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)
1A50 232 960 Omitted owing to length.
2A1 920 4A 6A 8A 10A 10B 12A ababbabababbababbabababbababbabababbababbabababb
3A1 080 6A 12A 15A 15B Omitted owing to length.
3B243 9A 9B 9C (ababb)3
4A96 8A 12A ababbabababbababbabababb
5A30 5B2 10A 10B 15A 15B Omitted owing to length.
5B30 5A2 10A 10B 15A 15B abbababbabababbabbabbababbabababbabb
6A24 12A abbababbabababbabbabababbabbababbabababbabbabababb
8A8 ababbabababb
9A27 9B4 9C2 ababbababbababbababb
9B27 9A2 9C4 ababb
9C27 9A4 9B2 ababbababb
10A10 10B3 abbababbabababbabb
10B10 10A3 (abbababbabababbabb)3
12A12 abbababbabababbabbabababb
15A15 15B2 abbababbabababbabbababbabababb
15B15 15A2 abbababbabababb
17A17 17B3 (abababb)3
17B17 17A3 abababb
19A19 19B2 ab
19B19 19A2 abab

Download words for class representatives.

Conjugacy classes of J3:2

Conjugacy class Centraliser order Power up Class rep(s)
1A100 465 920
2A3 840 4A 6A 8A 10A 12A 4B 8B 8C 12B 24A 24B
3A2 160 6A 12A 15A 12B 24A 24B
3B486 9A 9B 9C 6B 18A 18B 18C
4A192 8A 12A 8B 8C 24A 24B
5A30 10A 15A
6A48 12A 12B 24A 24B
8A16 cdcddcdcdcdcdcddcdcdcdcddcdcdcdd
9A54 9B4 9C2 18A 18B 18C
9B54 9A2 9C4 18A 18B 18C
9C54 9A4 9B2 18A 18B 18C
10A10 cdcdcdcddcdcdd
12A24 24A 24B
15A15 cdcdcdcdcddcdcdcdcddcdcdcdd
17A34 17B3 34A 34B
17B34 17A3 34A 34B
19A19 cdcdcddcdd
2B4 896 6B 18A 18B 18C 34A 34B
4B96 12B
6B18 18A 18B 18C
8B96 24A 24B
8C32 cdcddcddcdcdcdcdcddcdcdcdcddcdcdcdd
12B12 cdcdcdcdcddcdcdcdcdd
18A18 18B5 18C7
18B18 18A7 18C5 cdcdcdcdcdd
18C18 18A5 18B7
24A24 24B7 cd
24B24 24A7
34A34 34B3
34B34 34A3 cdcdcdd

Download words for class representatives.