Order = 86775571046077562880 = 221.33.5.7.113.23.29.31.37.43.
Mult = 1.
Out = 1.

Porting notes

Porting incomplete.

Standard generators

Type I standard generators of J4 are a, b where a is in class 2A, b is in class 4A, ab has order 37 and ababb has order 10.

Type II standard generators of J4 are x, y, t where x has order 2 (necessarily in class 2B), y has order 3, t has order 2 (necessarily in class 2A), (x, y) is a pair of Type I generators of M24, [t, x] has order 1 and [t, yxyxy2xy2xyxyxy] has order 1.

Black box algorithms

Finding generators

Group Algorithm File
J4 Download

Checking generators (semi-presentations)

Group Semi-presentation File
J4 〈〈 a, b | some conditions 〉〉 Download

Presentations

Group Presentation Link
J4 〈 x, y, t | x2 = y3 = (xy)23 = [x, y]12 = [x, yxy]5 = (xyxyxy−1)3(xyxy−1xy−1)3 = (xy(xyxy−1)3)4 = t2 = [t, x] = [t, yxy(xy−1)2(xy)3] = (ytyxy−1xyxy−1x)3 = ((yxyxyxy)3tt(xy)3y(xy)6y)2 = 1 〉 Details

Representations

Representations of J4

Maximal subgroups

Maximal subgroups of J4

Subgroup Order Index Programs/reps
211:M24 501 397 585 920 173 067 389 Program: Generators
21+12.3.M22:2 21 799 895 040 3 980 549 947 Program: Generators
210:L5(2) 10 239 344 640 8 474 719 242 Program: Generators
23+12.(S5 × L3(2)) 660 602 880 131 358 148 251 Program: Generators
U3(11):2 141 831 360 611 822 174 208 Program: Generators
M22:2 887 040 97 825 995 497 472 Program: Standard generators
111+2:(5 × 2S4) 319 440 271 649 045 348 352 Program: Generators
L2(32):5 163 680 530 153 782 050 816 Program: Standard generators
L2(32):2 12 144 7 145 550 975 467 520 Program: Generators
U3(3) 6 048 14 347 812 672 962 560 Program: Standard generators
29:28 = F812 812 106 866 466 805 514 240 Program: Generators
43:14 = F602 602 144 145 466 853 949 440 Program: Generators
37:12 = F444 444 195 440 475 329 003 520 Program: Generators

Conjugacy classes

Conjugacy classes of J4

Conjugacy class Centraliser order Power up Class rep(s)
1A 86 775 571 046 077 562 880
2A 21 799 895 040 4A 4B 6A 6B 8A 8B 8C 10A 12A 12B 14A 14B 16A 20A 20B 22A 24A 24B 30A 40A 40B 42A 42B 44A 66A 66B
2B 1 816 657 920 4C 6C 10B 12C 14C 14D 22B 28A 28B
3A 2 661 120 6A 6B 6C 12A 12B 12C 15A 21A 21B 24A 24B 30A 33A 33B 42A 42B 66A 66B
4A 5 406 720 8A 12A 20A 20B 40A 40B 44A
4B 98 304 8B 8C 12B 16A 24A 24B
4C 43 008 12C 28A 28B
5A 6 720 10A 10B 15A 20A 20B 30A 35A 35B 40A 40B
6A 2 661 120 30A 42A 42B 66A 66B
6B 2 304 12A 12B 24A 24B
6C 2 304 12C
7A 840 7B3 14A 14B 14C 14D 21A 21B 28A 28B 35A 35B 42A 42B
7B 840 7A3 14A 14B 14C 14D 21A 21B 28A 28B 35A 35B 42A 42B
8A 1 280 40A 40B
8B 768 24A 24B
8C 512 16A
10A 960 20A 20B 30A 40A 40B
10B 80 ababb
11A 31 944 22A 33A 33B 44A 66A 66B
11B 242 22B
12A 192 abababbabbabababbaababbabbabababb
12B 192 24A 24B
12C 48 babababb
14A 84 14B3 42A 42B
14B 84 14A3 42A 42B
14C 56 14D3 28A 28B
14D 56 14C3 28A 28B
15A 30 30A
16A 32 abababbabbabababb
20A 160 20B3 40A 40B
20B 160 20A3 40A 40B
21A 42 21B5 42A 42B
21B 42 21A5 42A 42B
22A 264 44A 66A 66B
22B 22 abbabababbabababbabbabababb
23A 23 aababbabbabababb
24A 48 24B5
24B 48 24A5
28A 28 28B3
28B 28 28A3
29A 29 abbaababbabbabababb
30A 30 abbabababbbabababb
31A 31 31B3 31C5
31B 31 31A5 31C3
31C 31 31A3 31B5
33A 66 33B5 66A 66B
33B 66 33A5 66A 66B
35A 35 35B3
35B 35 35A3
37A 37 37B3 37C2
37B 37 37A2 37C3
37C 37 37A3 37B2
40A 40 40B3
40B 40 40A3
42A 42 42B5
42B 42 42A5
43A 43 43B7 43C3
43B 43 43A3 43C7
43C 43 43A7 43B3
44A 44 babababbaababbabbabababb
66A 66 66B5
66B 66 66A5
24A–B abababb
28A–B ababbababbabbabababb
31A–C ababbbabababb
35A–B ababbabbabababbababbbabababb
37A–C ab
40A–B ababbabbabababbaababbabbabababb
42A–B babbabababbbabababb
43A–C ababbabbabababb
66A–B abbabababb

Download words for class representatives.