Order = 1255205709190661721292800 = 221.316.52.73.11.13.17.23.29.
Mult = 3.
Out = 2.

Porting notes

Porting incomplete.

Standard generators

Standard generators of Fi24' are a, b where a is in class 2A, b is in class 3E, ab has order 29 and abababb has order 33.

Standard generators of 3.Fi24' are preimages A, B where A has order 2 and AB has order 29.

Standard generators of Fi24':2 are c, d where c is in class 2C, d is in class 8D and cd has order 29.

Standard generators of 3.Fi24':2 are preimages C, D where D has order 29.

Black box algorithms

Finding generators

Group Algorithm File
Fi24' Download
Fi24':2 Download

Checking generators (semi-presentations)

Group Semi-presentation File
Fi24' 〈〈 a, b | o(a) = 2, o(b) = 3, o(ab) = 29, o(abababb) = 23, o(z) = 60, o(a(z30)abab) = 5; z = (ab)6b 〉〉 Download
Fi24':2 〈〈 c, d | o(c) = 2, o(d) = 8, o(cd) = 29, o(z) = 54, o(cz27) = 3, o(cdd) = 20; z = cdcd6 〉〉 Download

Representations

Representations of Fi24'

Representations of 3.Fi24'

Representations of Fi24':2

Representations of 3.Fi24':2

Conjugacy classes

Conjugacy classes of Fi24'

Conjugacy class Centraliser order Power up Class rep(s)
1A 1. 255 205 709 190 66e +24
2A 258 247 006 617 600 4B 6A 6C 6F 6G 10A 12D 12G 12L 14A 18B 18D 18F 20A 22A 26A 28A 30A 36D 42A 60A
2B 160 526 499 840 4A 4C 6B 6D 6E 6H 6I 6J 6K 8A 8B 8C 10B 12A 12B 12C 12E 12F 12H 12I 12J 12K 12M 14B 16A 18A 18C 18E 18G 18H 20B 24A 24B 24C 24D 24E 24F 24G 30B 36A 36B 36C 42B 42C
3A 44 569 618 329 600 6A 6D 12A 12C 12D 15A 21A 24A 24B 30A 39A 39B 42A 60A
3B 2 424 391 326 720 6C 6E 9A 9B 9C 9D 9E 12B 12G 12H 15B 18A 18B 18C 18D 18E 24E 27A 27B 27C 33A 33B 36A 36B 36C 36D 45A 45B
3C 14 285 134 080 6B 6F 6I 12E 12K 12L 15C 21C 21D 24F 24G 30B 42B 42C
3D 153 055 008 6G 6H 6J 9F 12F 12M 18F 18G 18H
3E 38 211 264 6K 12I 12J 21B 24C 24D 39C 39D
4A 39 813 120 8A 12A 12B 12C 12E 12F 12I 12J 20B 24A 24B 24C 24D 36A 36B 36C
4B 5 806 080 12D 12G 12L 20A 28A 36D 60A
4C 294 912 8B 8C 12H 12K 12M 16A 24E 24F 24G
5A 907 200 10A 10B 15A 15B 15C 20A 20B 30A 30B 35A 45A 45B 60A
6A 78 382 080 12D 30A 42A 60A
6B 19 595 520 30B 42B 42C
6C 10 077 696 12G 18B 18D 36D
6D 2 985 984 12A 12C 24A 24B
6E 2 239 488 12B 12H 18A 18C 18E 24E 36A 36B 36C
6F 559 872 12L
6G 69 984 18F
6H 69 984 18G 18H
6I 62 208 12E 12K 24F 24G
6J 23 328 12F 12M
6K 5 184 12I 12J 24C 24D
7A 17 640 14A 21A 21B 28A 35A 42A
7B 2 058 14B 21C 21D 42B 42C
8A 2 304 24A 24B 24C 24D
8B 1 536 16A 24E
8C 768 24F 24G
9A 472 392 18C 27A 36C
9B 262 440 18D 36D 45A 45B
9C 209 952 18A 18B 36A 36B
9D 59 049 27B 27C
9E 13 122 18E
9F 972 18F 18G 18H
10A 2 400 20A 30A 60A
10B 960 20B 30B
11A 132 22A 33A 33B
12A 124 416 24A
12B 31 104 36A 36B 36C
12C 13 824 24B
12D 8 640 60A
12E 5 184 Omitted owing to length.
12F 3 888 Omitted owing to length.
12G 2 592 36D
12H 1 152 24E
12I 864 24C
12J 864 24D
12K 576 24F 24G
12L 432 ababababbababababababb
12M 144 ababababbababababababbabababbababababbabababababbababababababb
13A 234 26A 39A 39B 39C 39D
14A 168 28A 42A
14B 42 42B 42C
15A 5 400 30A 60A
15B 405 45A 45B
15C 270 30B
16A 32 abababababbabababababbabababbababababb
17A 17 abababbababababb
18A 2 592 36A 36B
18B 1 296 Omitted owing to length.
18C 648 36C
18D 648 36D
18E 162 abababbababababbabababababbababababababb
18F 108 abababababbabababbababababbabababbababababababb
18G 108 18H5
18H 108 18G5
20A 120 60A
20B 80 ababababbababababbababababbabababababb
21A 252 42A
21B 63 abababbababababbabababbabababababb
21C 42 21D2 42B 42C
21D 42 21C2 42B 42C
22A 44 ababababababbababababbabababababb
23A 23 23B5
23B 23 23A5
24A 288 Omitted owing to length.
24B 288 ababababbabababbababababbabbabababbababababb
24C 72
24D 72
24E 48 babababbababababbabababbabababababb
24F 48 24G5
24G 48 24F5
26A 26 abababbababababababb
27A 81 ababbababababbababababababb
27B 81
27C 81
28A 28 ababababbabababababb
29A 29 29B2
29B 29 29A2
30A 120 60A
30B 30 abababababbabababbababababb
33A 33 33B5
33B 33 33A5
35A 35 abababbabababababb
36A 432
36B 432
36C 108 abababbabababababbabababbababababbabababbabababababb
36D 36 abababababbababababababb
39A 117 39B2
39B 117 39A2
39C 117 39D2
39D 117 39C2
42A 84 abababbababababbabababbababababbabababbabababababb
42B 42 42C11
42C 42 42B11
45A 45 45B2
45B 45 45A2
60A 60 ababababababb
18G-H ababababababbababababababbababababbabababababb
23A-B abababbababababbabbabababbababababb
24C-D abababbabababababbababababbababababbabababababb
24F-G ababababababbabababbababababababb
27B-C abababababb
29A-B ab
33A-B abababb
36A-B ababababb
39A-B abbabababbababababbabababababbababababababb
39C-D abbababababbababababbabababababb
42B-C ababababbababababbabababababb
45A-B abbabababbababababb

Download words for class representatives.

Conjugacy classes of Fi24':2

Conjugacy class Centraliser order Power up Class rep(s)
1A 2 510 411 418 381 323 442 585 600
2A 516 494 013 235 200 4B 4B 6A 6C 6F 6G 10A 12D 12G 12K 14A 18B 18D 18F 20A 22A 26A 28A 30A 36C 42A 60A 4D 4E 12M 12N 12O 12Q 12R 12U 12V 12W 20C 20D 28B 36D 36E 36G 60B 84A
2B 321 052 999 680 4A 4A 4C 4C 6B 6D 6E 6H 6I 6J 6K 8A 8B 8C 10B 12A 12B 12C 12E 12F 12H 12I 12J 12L 14B 16A 18A 18C 18E 18G 20B 24A 24B 24C 24D 24E 30B 36A 36B 42B 4F 4G 8D 8E 8F 12P 12S 12T 12X 12Y 12Z 12A1 16B 24F 24G 24H 24I 28C 28D 36F 40A
3A 89 139 236 659 200 6A 6D 12A 12C 12D 15A 21A 24A 24B 30A 39A 39B 42A 60A 6L 6P 12M 12N 12R 12S 24F 24G 30C 30D 42C 60B 78A 78B 84A
3B 4 848 782 653 440 6C 6E 9A 9B 9C 9D 9E 12B 12G 12H 15B 18A 18B 18C 18D 18E 24D 27A 27B 33A 33B 36A 36B 36C 45A 6M 6N 6Q 12O 12P 12T 18H 18I 18J 18K 18L 18M 18N 18O 18P 24I 30G 36D 36E 36F 54A 66A 66B
3C 28 570 268 160 6B 6F 6I 12E 12J 12K 15C 21C 24E 30B 42B 6O 6S 6T 12Q 12W 12X 24H 30E 30F
3D 306 110 016 6G 6H 6J 9F 12F 12L 18F 18G 6R 6U 12U 12V 12Y 12Z 18Q 36G
3E 76 422 528 6K 12I 21B 24C 39C 6V 12A1 42D
4A 79 626 240 8A 12A 12B 12C 12E 12F 12I 20B 24A 24B 24C 36A 36B 8D 8E 24F 24G 24H 40A
4B 11 612 160 12D 12G 12K 20A 28A 36C 60A
4C 589 824 8B 8C 12H 12J 12L 16A 24D 24E 8F 16B 24I
5A 1 814 400 10A 10B 15A 15B 15C 20A 20B 30A 30B 35A 45A 60A 10C 10D 20C 20D 30C 30D 30E 30F 30G 40A 60B 70A
6A 156 764 160 12D 30A 42A 60A 12M 12N 12R 60B 84A
6B 39 191 040 30B 42B
6C 20 155 392 12G 18B 18D 36C 12O 36D 36E
6D 5 971 968 12A 12C 24A 24B 12S 24F 24G
6E 4 478 976 12B 12H 18A 18C 18E 24D 36A 36B 12P 12T 24I 36F
6F 1 119 744 12K 12Q 12W
6G 139 968 18F 12U 12V 36G
6H 139 968 18G 12Y
6I 124 416 12E 12J 24E 12X 24H
6J 46 656 12F 12L 12Z
6K 10 368 12I 24C 12A1
7A 35 280 14A 21A 21B 28A 35A 42A 14C 14D 28B 42C 42D 70A 84A
7B 4 116 14B 21C 42B 28C 28D
8A 4 608 24A 24B 24C
8B 3 072 16A 24D 16B
8C 1 536 24E
9A 944 784 18C 27A 36B 18H 18K 36F 54A
9B 524 880 18D 36C 45A 18I 18M 36E
9C 419 904 18A 18B 36A 18J 18N 36D
9D 118 098 27B 18O
9E 26 244 18E 18L 18P
9F 1 944 18F 18G 18Q 36G
10A 4 800 20A 30A 60A 20C 20D 60B
10B 1 920 20B 30B 40A
11A 264 22A 33A 33B 22B 22C 66A 66B
12A 248 832 24A 24F
12B 62 208 36A 36B
12C 27 648 24B 24G
12D 17 280 60A
12E 10 368 24H
12F 7 776
12G 5 184 36C
12H 2 304 24D 24I
12I 864 24C
12J 1 152 24E
12K 864
12L 288
13A 468 26A 39A 39B 39C 26B 26C 78A 78B
14A 336 28A 42A 28B 84A
14B 84 42B 28C 28D
15A 10 800 30A 60A 30C 30D 60B
15B 810 45A 30G
15C 540 30B 30E 30F
16A 64
17A 34 34A
18A 5 184 36A
18B 2 592 36D
18C 1 296 36B 36F
18D 1 296 36C 36E
18E 324
18F 216 36G
18G 108
20A 240 60A
20B 160 40A
21A 504 42A 42C 84A
21B 126 42D
21C 42 42B
22A 88
23A 46 23B5 46A 46B
23B 46 23A5 46A 46B
24A 576
24B 576
24C 72
24D 96
24E 48
26A 52
27A 162 54A
27B 81
28A 56
29A 29
30A 240 60A 60B
30B 60
33A 66 33B5 66A 66B
33B 66 33A5 66A 66B
35A 70 70A
36A 432
36B 216
36C 72
39A 234 39B2 78A 78B
39B 234 39A2 78A 78B
39C 117
42A 168 84A
42B 42
45A 45
60A 120
2C 8 178 940 946 586 009 600 6L 6M 6O 6R 10C 14C 18H 18I 18J 18L 18Q 22C 26B 26C 30C 30E 34A 42C 46A 46B 54A 70A 78A 78B
2D 441 447 874 560 6N 6P 6Q 6S 6T 6U 6V 10D 14D 18K 18M 18N 18O 18P 22B 30D 30F 30G 42D 66A 66B
4D 4 180 377 600 12M 12N 12O 12Q 12U 20C 28B 36D 36E 60B 84A
4E 13 271 040 12R 12V 12W 20D 36G
4F 2 654 208 12P 12S 12T 12X 12Z 36F
4G 1 548 288 12Y 12A1 28C 28D
6L 27 512 110 080 30C 42C 78A 78B
6M 3 265 173 504 18H 18I 18J 18L 54A
6N 82 114 560 30G 66A 66B
6O 75 582 720 30E
6P 3 732 480 30D
6Q 2 239 488 18K 18M 18N 18O 18P
6R 1 889 568 18Q
6S 933 120 30F
6T 93 312
6U 23 328
6V 9 072 42D
8D 15 360 24G 40A
8E 9 216 24F 24H
8F 3 072 24I
10C 50 400 30C 30E 70A
10D 1 440 30D 30F 30G
12M 622 080 60B
12N 145 152 84A
12O 46 656 36D 36E
12P 20 736 36F
12Q 15 552
12R 13 824
12S 6 912
12T 6 912
12U 2 592
12V 2 592 36G
12W 1 728
12X 1 728
12Y 432
12Z 432
12A1 144
14C 1 680 42C 70A
14D 336 42D
16B 64
18H 34 992 54A
18I 11 664
18J 11 664
18K 3 888
18L 2 916
18M 1 296
18N 1 296
18O 486
18P 324
18Q 324
20C 2 400 60B
20D 160
22B 264 66A 66B
22C 88
24F 576
24G 192
24H 144
24I 96
26B 156 26C5 78A 78B
26C 156 26B5 78A 78B
28B 168 84A
28C 28 28D5
28D 28 28C5
30C 720
30D 720
30E 180
30F 180
30G 90
34A 34
36D 216
36E 216
36F 72
36G 36
40A 40
42C 84
42D 42
46A 46 46B5
46B 46 46A5
54A 54
60B 120
66A 66 66B5
66B 66 66A5
70A 70
78A 78 78B5
78B 78 78A5
84A 84

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