Order = 4089470473293004800 = 218.313.52.7.11.13.17.23.
Mult = 1.
Out = 1.

## Porting notes

Fully copied from version 2. 30/7/06.

## Standard generators

Standard generators of Fi23 are a, b where a is in class 2B, b is in class 3D and ab has order 28.

## Black box algorithms

### Finding generators

Group Algorithm File

### Checking generators (semi-presentations)

Group Semi-presentation File
Fi23 〈〈 a, b | o(a) = 2, o(b) = 3, o(ab) = 28, o(abb(ab)14) = 5 〉〉 Download

## Maximal subgroups

### Maximal subgroups of Fi23

Subgroup Order Index Programs/reps
2.Fi22 129 123 503 308 800 31 671 Program: Generators
Program: Generators
O8+(3):S3 29 713 078 886 400 137 632 Program: Generators
22.U6(2).2 73 574 645 760 55 582 605 Program: Generators
S8(2) 47 377 612 800 86 316 516 Program: Generators
O7(3) × S3 27 512 110 080 148 642 560 Program: Generators
211.M23 20 891 566 080 195 747 435 Program: Generators
31+8.21+6.31+2.2S4 3 265 173 504 1 252 451 200
[310].(L3(3) × 2) 663 238 368 6 165 913 600
S12 479 001 600 8 537 488 128 Program: Generators
(22 × 21+8).(3 × U4(2)).2 318 504 960 12 839 581 755 Program: Generators
26+8:(A7 × S3) 247 726 080 16 508 033 685
S6(2) × S4 34 836 480 117 390 461 760
S4(4):4 3 916 800 1 044 084 577 536 Program: Generators
L2(23) 6 072 673 496 454 758 400 Program: Generators

## Conjugacy classes

### Conjugacy classes of Fi23

Conjugacy class Centraliser order Power up Class rep(s)
1A 4 089 470 473 293 004 800
2A 129 123 503 308 800 6A 6B 6E 6J 10A 14A 18A 18B 18H 22B 22C 26A 26B 30B 42A
2B 73 574 645 760 4A 4C 6C 6D 6K 10B 12A 12D 12I 12J 12N 14B 18E 18F 20A 20B 22A 28A 30A 36B 60A
2C 318 504 960 4B 4D 6F 6G 6H 6I 6L 6M 6N 6O 8A 8B 8C 10C 12B 12C 12E 12F 12G 12H 12K 12L 12M 12O 16A 16B 18C 18D 18G 24A 24B 24C 30C 36A
3A 13 756 055 040 6A 6C 6I 12A 12C 12G 12I 15A 21A 24A 24C 30A 30B 39A 39B 42A 60A
3B 1 632 586 752 6B 6D 6F 6H 9A 9B 9C 9D 12B 12D 12E 12F 12K 18A 18B 18C 18D 18E 18F 18G 24B 27A 36A 36B
3C 37 791 360 6E 6G 6K 6L 6M 12H 12J 12L 12N 15B 30C
3D 944 784 6J 6N 6O 9E 12M 12O 18H
4A 5 806 080 12A 12D 12J 20A 28A 36B 60A
4B 331 776 8A 8B 12B 12C 12E 12G 12H 12M 24A 24C 36A
4C 92 160 12I 12N 20B
4D 36 864 8C 12F 12K 12L 12O 16A 16B 24B
5A 25 200 10A 10B 10C 15A 15B 20A 20B 30A 30B 30C 35A 60A
6A 39 191 040 30B 42A
6B 5 038 848 18A 18B
6C 622 080 12A 12I 30A 60A
6D 373 248 12D 18E 18F 36B
6E 279 936 Omitted owing to length.
6F 248 832 12B 12F 24B
6G 155 520 30C
6H 62 208 12E 12K 18C 18D 18G 36A
6I 41 472 12C 12G 24A 24C
6J 34 992 18H Omitted owing to length.
6K 15 552 12J 12N
6L 10 368 12H 12L
6M 5 184 Omitted owing to length.
6N 3 888 abaabababbabababbababababbababababbababababbabababbababababb
6O 1 296 12M 12O
7A 840 14A 14B 21A 28A 35A 42A
8A 384 24C
8B 384 24A
8C 384 16A 16B 24B
9A 17 496 18D 27A 36A
9B 5 832 18B 18E 36B
9C 5 832 18A 18C 18F
9D 1 458 18G
9E 162 18H
10A 1 200 30B
10B 240 20A 20B 30A 60A
10C 240 30C
11A 44 22A 22B 22C
12A 8 640 60A
12B 3 456 ababbabababbabababbabababbababababbababababbabababbababababb
12C 3 456 24A
12D 2 592 36B
12E 2 592 36A
12F 1 152 24B
12G 1 152 24C
12H 864 abababbabababbabababbababababb
12I 576 Omitted owing to length.
12J 432 ababbababababbabababbababababbababababbabababbababababb
12K 288 ababababb
12L 288 Omitted owing to length.
12M 216 ababababbabababbababababb
12N 144 Omitted owing to length.
12O 72 ababababbaabababbabababbababababb
13A 78 13B2 26A 26B 39A 39B
13B 78 13A2 26A 26B 39A 39B
14A 84 42A
14B 56 28A
15A 360 30A 30B 60A
15B 90 30C
16A 32 16B5
16B 32 16A5
17A 17 ababbabababbabababbababababb
18A 648 Omitted owing to length.
18B 324 babababbabababbabababbababababbababababbabababbababababb
18C 216 Omitted owing to length.
18D 216 36A
18E 216 36B
18F 216 abababababbabababbabababbababababb
18G 54 ababbaabababbabababbababababb
18H 54 ababababbabababbababababbababababbabababbabababbababababb
20A 120 60A
20B 40 abababbabababbababababbababababbabababbababababb
21A 42 42A
22A 44 aabababbabababbababababb
22B 44 22C7
22C 44 22B7
23A 23 23B5
23B 23 23A5
24A 48 Omitted owing to length.
24B 48 abababbabababbabababbababababbababababbabababbababababb
24C 48 aabababbabababbabababbababababb
26A 26 26B5
26B 26 26A5
27A 27 Omitted owing to length.
28A 28 ab
30A 120 60A
30B 60 ababababbabababbabababbababababb
30C 30 ababababbababababbabababbababababb
35A 35 abbababbabababbabababbababababb
36A 36 abababbabababbababababbabababbabababbabababbababababb
36B 36 Omitted owing to length.
39A 39 39B2
39B 39 39A2
42A 42 Omitted owing to length.
60A 60 Omitted owing to length.
16A-B aabababbabababbababababbababababbababababbabababbababababb
22B-C aabababbabababbababababbabbababbabababbabababbababababb
23A-B ababbabababbabababbababababbabababbabababbabababbababababb
26A-B abababbababababb
39A-B abababbabababbababababb