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

### Checking generators (semi-presentations)

Group Semi-presentation File

## Presentations

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

## Maximal subgroups

### Maximal subgroups of J4

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

## Conjugacy classes

### Conjugacy classes of J4

Conjugacy class Centraliser order Power up Class rep(s)
1A86 775 571 046 077 562 880
2A21 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
2B1 816 657 920 4C 6C 10B 12C 14C 14D 22B 28A 28B
3A2 661 120 6A 6B 6C 12A 12B 12C 15A 21A 21B 24A 24B 30A 33A 33B 42A 42B 66A 66B
4A5 406 720 8A 12A 20A 20B 40A 40B 44A
4B98 304 8B 8C 12B 16A 24A 24B
4C43 008 12C 28A 28B
5A6 720 10A 10B 15A 20A 20B 30A 35A 35B 40A 40B
6A2 661 120 30A 42A 42B 66A 66B
6B2 304 12A 12B 24A 24B
6C2 304 12C
7A840 7B3 14A 14B 14C 14D 21A 21B 28A 28B 35A 35B 42A 42B
7B840 7A3 14A 14B 14C 14D 21A 21B 28A 28B 35A 35B 42A 42B
8A1 280 40A 40B
8B768 24A 24B
8C512 16A
10A960 20A 20B 30A 40A 40B
10B80 ababb
11A31 944 22A 33A 33B 44A 66A 66B
11B242 22B
12A192 abababbabbabababbaababbabbabababb
12B192 24A 24B
12C48 babababb
14A84 14B3 42A 42B
14B84 14A3 42A 42B
14C56 14D3 28A 28B
14D56 14C3 28A 28B
15A30 30A
16A32 abababbabbabababb
20A160 20B3 40A 40B
20B160 20A3 40A 40B
21A42 21B5 42A 42B
21B42 21A5 42A 42B
22A264 44A 66A 66B
22B22 abbabababbabababbabbabababb
23A23 aababbabbabababb
24A48 24B5
24B48 24A5
28A28 28B3
28B28 28A3
29A29 abbaababbabbabababb
30A30 abbabababbbabababb
31A31 31B3 31C5
31B31 31A5 31C3
31C31 31A3 31B5
33A66 33B5 66A 66B
33B66 33A5 66A 66B
35A35 35B3
35B35 35A3
37A37 37B3 37C2
37B37 37A2 37C3
37C37 37A3 37B2
40A40 40B3
40B40 40A3
42A42 42B5
42B42 42A5
43A43 43B7 43C3
43B43 43A3 43C7
43C43 43A7 43B3
44A44 babababbaababbabbabababb
66A66 66B5
66B66 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