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'
Fi24':2

### 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'

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
306936StdDetails
• Matrix representations
Char Ring Dimension ID Generators Description Link
2GF(2)3774StdDetails
3GF(3)781StdDetails

### Representations of 3.Fi24'

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
920808StdDetails
• Matrix representations
Char Ring Dimension ID Generators Description Link
2GF(4)783StdDetails

### Representations of Fi24':2

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
306936StdDetails
• Matrix representations
Char Ring Dimension ID Generators Description Link
3GF(3)781StdDetails

### Representations of 3.Fi24':2

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
920808StdDetails
• Matrix representations
Char Ring Dimension ID Generators Description Link
2GF(2)1566StdDetails

## Conjugacy classes

### Conjugacy classes of Fi24'

Conjugacy class Centraliser order Power up Class rep(s)
1A1. 255 205 709 190 66e +24
2A258 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
2B160 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
3A44 569 618 329 600 6A 6D 12A 12C 12D 15A 21A 24A 24B 30A 39A 39B 42A 60A
3B2 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
3C14 285 134 080 6B 6F 6I 12E 12K 12L 15C 21C 21D 24F 24G 30B 42B 42C
3D153 055 008 6G 6H 6J 9F 12F 12M 18F 18G 18H
3E38 211 264 6K 12I 12J 21B 24C 24D 39C 39D
4A39 813 120 8A 12A 12B 12C 12E 12F 12I 12J 20B 24A 24B 24C 24D 36A 36B 36C
4B5 806 080 12D 12G 12L 20A 28A 36D 60A
4C294 912 8B 8C 12H 12K 12M 16A 24E 24F 24G
5A907 200 10A 10B 15A 15B 15C 20A 20B 30A 30B 35A 45A 45B 60A
6A78 382 080 12D 30A 42A 60A
6B19 595 520 30B 42B 42C
6C10 077 696 12G 18B 18D 36D
6D2 985 984 12A 12C 24A 24B
6E2 239 488 12B 12H 18A 18C 18E 24E 36A 36B 36C
6F559 872 12L
6G69 984 18F
6H69 984 18G 18H
6I62 208 12E 12K 24F 24G
6J23 328 12F 12M
6K5 184 12I 12J 24C 24D
7A17 640 14A 21A 21B 28A 35A 42A
7B2 058 14B 21C 21D 42B 42C
8A2 304 24A 24B 24C 24D
8B1 536 16A 24E
8C768 24F 24G
9A472 392 18C 27A 36C
9B262 440 18D 36D 45A 45B
9C209 952 18A 18B 36A 36B
9D59 049 27B 27C
9E13 122 18E
9F972 18F 18G 18H
10A2 400 20A 30A 60A
10B960 20B 30B
11A132 22A 33A 33B
12A124 416 24A
12B31 104 36A 36B 36C
12C13 824 24B
12D8 640 60A
12E5 184 Omitted owing to length.
12F3 888 Omitted owing to length.
12G2 592 36D
12H1 152 24E
12I864 24C
12J864 24D
12K576 24F 24G
12L432 ababababbababababababb
12M144 ababababbababababababbabababbababababbabababababbababababababb
13A234 26A 39A 39B 39C 39D
14A168 28A 42A
14B42 42B 42C
15A5 400 30A 60A
15B405 45A 45B
15C270 30B
16A32 abababababbabababababbabababbababababb
17A17 abababbababababb
18A2 592 36A 36B
18B1 296 Omitted owing to length.
18C648 36C
18D648 36D
18E162 abababbababababbabababababbababababababb
18F108 abababababbabababbababababbabababbababababababb
18G108 18H5
18H108 18G5
20A120 60A
20B80 ababababbababababbababababbabababababb
21A252 42A
21B63 abababbababababbabababbabababababb
21C42 21D2 42B 42C
21D42 21C2 42B 42C
22A44 ababababababbababababbabababababb
23A23 23B5
23B23 23A5
24A288 Omitted owing to length.
24B288 ababababbabababbababababbabbabababbababababb
24C72
24D72
24E48 babababbababababbabababbabababababb
24F48 24G5
24G48 24F5
26A26 abababbababababababb
27A81 ababbababababbababababababb
27B81
27C81
28A28 ababababbabababababb
29A29 29B2
29B29 29A2
30A120 60A
30B30 abababababbabababbababababb
33A33 33B5
33B33 33A5
35A35 abababbabababababb
36A432
36B432
36C108 abababbabababababbabababbababababbabababbabababababb
36D36 abababababbababababababb
39A117 39B2
39B117 39A2
39C117 39D2
39D117 39C2
42A84 abababbababababbabababbababababbabababbabababababb
42B42 42C11
42C42 42B11
45A45 45B2
45B45 45A2
60A60 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

### Conjugacy classes of Fi24':2

Conjugacy class Centraliser order Power up Class rep(s)
1A2 510 411 418 381 323 442 585 600
2A516 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
2B321 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
3A89 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
3B4 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
3C28 570 268 160 6B 6F 6I 12E 12J 12K 15C 21C 24E 30B 42B 6O 6S 6T 12Q 12W 12X 24H 30E 30F
3D306 110 016 6G 6H 6J 9F 12F 12L 18F 18G 6R 6U 12U 12V 12Y 12Z 18Q 36G
3E76 422 528 6K 12I 21B 24C 39C 6V 12A1 42D
4A79 626 240 8A 12A 12B 12C 12E 12F 12I 20B 24A 24B 24C 36A 36B 8D 8E 24F 24G 24H 40A
4B11 612 160 12D 12G 12K 20A 28A 36C 60A
4C589 824 8B 8C 12H 12J 12L 16A 24D 24E 8F 16B 24I
5A1 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
6A156 764 160 12D 30A 42A 60A 12M 12N 12R 60B 84A
6B39 191 040 30B 42B
6C20 155 392 12G 18B 18D 36C 12O 36D 36E
6D5 971 968 12A 12C 24A 24B 12S 24F 24G
6E4 478 976 12B 12H 18A 18C 18E 24D 36A 36B 12P 12T 24I 36F
6F1 119 744 12K 12Q 12W
6G139 968 18F 12U 12V 36G
6H139 968 18G 12Y
6I124 416 12E 12J 24E 12X 24H
6J46 656 12F 12L 12Z
6K10 368 12I 24C 12A1
7A35 280 14A 21A 21B 28A 35A 42A 14C 14D 28B 42C 42D 70A 84A
7B4 116 14B 21C 42B 28C 28D
8A4 608 24A 24B 24C
8B3 072 16A 24D 16B
8C1 536 24E
9A944 784 18C 27A 36B 18H 18K 36F 54A
9B524 880 18D 36C 45A 18I 18M 36E
9C419 904 18A 18B 36A 18J 18N 36D
9D118 098 27B 18O
9E26 244 18E 18L 18P
9F1 944 18F 18G 18Q 36G
10A4 800 20A 30A 60A 20C 20D 60B
10B1 920 20B 30B 40A
11A264 22A 33A 33B 22B 22C 66A 66B
12A248 832 24A 24F
12B62 208 36A 36B
12C27 648 24B 24G
12D17 280 60A
12E10 368 24H
12F7 776
12G5 184 36C
12H2 304 24D 24I
12I864 24C
12J1 152 24E
12K864
12L288
13A468 26A 39A 39B 39C 26B 26C 78A 78B
14A336 28A 42A 28B 84A
14B84 42B 28C 28D
15A10 800 30A 60A 30C 30D 60B
15B810 45A 30G
15C540 30B 30E 30F
16A64
17A34 34A
18A5 184 36A
18B2 592 36D
18C1 296 36B 36F
18D1 296 36C 36E
18E324
18F216 36G
18G108
20A240 60A
20B160 40A
21A504 42A 42C 84A
21B126 42D
21C42 42B
22A88
23A46 23B5 46A 46B
23B46 23A5 46A 46B
24A576
24B576
24C72
24D96
24E48
26A52
27A162 54A
27B81
28A56
29A29
30A240 60A 60B
30B60
33A66 33B5 66A 66B
33B66 33A5 66A 66B
35A70 70A
36A432
36B216
36C72
39A234 39B2 78A 78B
39B234 39A2 78A 78B
39C117
42A168 84A
42B42
45A45
60A120
2C8 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
2D441 447 874 560 6N 6P 6Q 6S 6T 6U 6V 10D 14D 18K 18M 18N 18O 18P 22B 30D 30F 30G 42D 66A 66B
4D4 180 377 600 12M 12N 12O 12Q 12U 20C 28B 36D 36E 60B 84A
4E13 271 040 12R 12V 12W 20D 36G
4F2 654 208 12P 12S 12T 12X 12Z 36F
4G1 548 288 12Y 12A1 28C 28D
6L27 512 110 080 30C 42C 78A 78B
6M3 265 173 504 18H 18I 18J 18L 54A
6N82 114 560 30G 66A 66B
6O75 582 720 30E
6P3 732 480 30D
6Q2 239 488 18K 18M 18N 18O 18P
6R1 889 568 18Q
6S933 120 30F
6T93 312
6U23 328
6V9 072 42D
8D15 360 24G 40A
8E9 216 24F 24H
8F3 072 24I
10C50 400 30C 30E 70A
10D1 440 30D 30F 30G
12M622 080 60B
12N145 152 84A
12O46 656 36D 36E
12P20 736 36F
12Q15 552
12R13 824
12S6 912
12T6 912
12U2 592
12V2 592 36G
12W1 728
12X1 728
12Y432
12Z432
12A1144
14C1 680 42C 70A
14D336 42D
16B64
18H34 992 54A
18I11 664
18J11 664
18K3 888
18L2 916
18M1 296
18N1 296
18O486
18P324
18Q324
20C2 400 60B
20D160
22B264 66A 66B
22C88
24F576
24G192
24H144
24I96
26B156 26C5 78A 78B
26C156 26B5 78A 78B
28B168 84A
28C28 28D5
28D28 28C5
30C720
30D720
30E180
30F180
30G90
34A34
36D216
36E216
36F72
36G36
40A40
42C84
42D42
46A46 46B5
46B46 46A5
54A54
60B120
66A66 66B5
66B66 66A5
70A70
78A78 78B5
78B78 78A5
84A84

Download words (if any exist) for class representatives.