Order = 64561751654400 = 217.39.52.7.11.13.
Mult = 6.
Out = 2.

## Porting notes

Porting incomplete.

## Standard generators

Standard generators of Fi22 are a, b where a is in class 2A, b has order 13, ab has order 11 and ababababbababbabb has order 12.

Standard generators of 2.Fi22 are preimages A, B where B has order 13 and AB has order 11.

Standard generators of 3.Fi22 are preimages A, B where A has order 2 and B has order 13.

Standard generators of Fi22:2 are c, d where c is in class 2A, d is in class 18E and cd has order 42.

Standard generators of 2.Fi22:2 are preimages C, D where CDCDCDCDCDD has order 30. Alternatively: D is in class +18E and CD is in class +42A.

Standard generators of 3.Fi22:2 are preimages C, D where C has order 2.

Standard generators of 6.Fi22 are A, B where A is in class ...? and B is in class ...?.

## Black box algorithms

### Finding generators

Group Algorithm File

### Checking generators (semi-presentations)

Group Semi-presentation File
Fi22 〈〈 a, b | o(a) = 2, o(b) = 13, o(ab) = 11, o(ababababbababbabb) = 12, o(z) = 30, o(az15) = 3; z = ababbabb 〉〉 Download
Fi22:2 〈〈 c, d | o(c) = 2, o(d) = 18, o(cd) = 42, o(z) = 22, o(cz11) = 3, o((d9)cddd(cd)21) = 3; z = cdcd5cd4 〉〉 Download

## Presentations

Fi22 a, b | a2 = b13 = (ab)11 = (ab2)21 = [a, b]3 = [a, b2]3 = [a, b3]3 = [a, b4]2 = [a, b5]3 = [a, bab2]3 = [a, b−1ab−2]2 = [a, bab5]2 = [a, b2ab5]2 = 1 〉 Details
2.Fi22 A, B | A2 = B13 = (AB)11 = [A,B]3 = [A,B2]3 = [A,B3]3 = [A,B4]2 = [A,B5]3 = [A,BAB2]3 = [A,B−1AB−2]2, = [A,BAB5]2, = [A,B−1AB−5]3 = [A,B2AB5]2 = 1 〉 Details
Fi22:2 c, d | c2 = d18 = [c, d]3 = [c, d2]3 = [c, d3]3 = [c, d4]3 = [c, d5]3 = [c, d6]2 = [c, d7]2 = [c, d8]3 = (cd9)4 = [c, dcdcd−2cd] = [c, d2cd2cd−4cd2] = ((cd3)4cd−4)2 = (cd4cd5cd5)5 = (cdcd−3)8 = 1 〉 Details
3.Fi22:2 C, D | C2 = D18 = [C,D6]2 = [C,D7]2 = (CD9)4 = [C,D]3 = [C,D2]3 = [C,D3]3 = [C,D4]3 = [C,D5]3 = [C,D8]3 = [C,DCDCD−2CD] = [C,D2CD2CD−4CD2] = ((CD3)4CD−4)2 = (CD4CD5CD5)5 = 1 〉 Details

## Representations

### Representations of Fi22

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
3510 Std Details
14080 b Std Details
61776 Std Details
142155 Std Details
694980 Std Details
• Matrix representations
Char Ring Dimension ID Generators Description Link
0 Z 78 Std Details
Char Ring Dimension ID Generators Description Link
2 GF(2) 78 Std Details
2 GF(2) 350 Std Details
2 GF(2) 572 Std Details
Char Ring Dimension ID Generators Description Link
3 GF(3) 77 Std Details
3 GF(3) 351 Std Details
3 GF(3) 924 Std Details
Char Ring Dimension ID Generators Description Link
5 GF(5) 78 Std Details
5 GF(5) 428 Std Details
Char Ring Dimension ID Generators Description Link
7 GF(7) 78 Std Details
7 GF(7) 429 Std Details
Char Ring Dimension ID Generators Description Link
11 GF(11) 78 Std Details
11 GF(11) 429 Std Details
Char Ring Dimension ID Generators Description Link
13 GF(13) 78 Std Details
13 GF(13) 429 Std Details

### Representations of 3.Fi22

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
185328 Std Details
• Matrix representations
Char Ring Dimension ID Generators Description Link
2 GF(4) 27 a Std Details
7 GF(7) 351 a Std Details

### Representations of Fi22:2

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
3510 Std Details
• Matrix representations
Char Ring Dimension ID Generators Description Link
0 Z 78 Std Details
Char Ring Dimension ID Generators Description Link
2 GF(2) 78 Std Details
2 GF(2) 350 Std Details
2 GF(2) 572 Std Details
2 GF(2) 1352 Std Details
Char Ring Dimension ID Generators Description Link
3 GF(3) 77 a Std Details
3 GF(3) 351 Std Details
3 GF(3) 924 Std Details
Char Ring Dimension ID Generators Description Link
5 GF(5) 78 a Std Details
5 GF(5) 428 a Std Details
Char Ring Dimension ID Generators Description Link
7 GF(7) 78 a Std Details
7 GF(7) 429 a Std Details
Char Ring Dimension ID Generators Description Link
11 GF(11) 78 a Std Details
11 GF(11) 429 a Std Details
Char Ring Dimension ID Generators Description Link
13 GF(13) 78 a Std Details
13 GF(13) 429 a Std Details

### Representations of 2.Fi22:2

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
56320 Std Details
• Matrix representations
Char Ring Dimension ID Generators Description Link
3 GF(3) 352 Std Details

### Representations of 3.Fi22:2

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
185328 Std Details
• Matrix representations
Char Ring Dimension ID Generators Description Link
2 GF(2) 54 Std Details
7 GF(7) 702 Std Details

## Maximal subgroups

### Maximal subgroups of Fi22

Subgroup Order Index Programs/reps
2.U6(2) 18 393 661 440 3 510 Program: Generators
O7(3) 4 585 351 680 14 080 Program: Generators
O7(3) 4 585 351 680 14 080 Program: Generators
Program: Generators
O8+(2):S3 1 045 094 400 61 776 Program: Generators
210:M22 454 164 480 142 155 Program: Generators
26:S6(2) 92 897 280 694 980 Program: Generators
(2 × 21+8):(U4(2):2) = 2.21+8:(U4(2):2) 53 084 160 1 216 215 Program: Generators
U4(3):2 × S3 39 191 040 1 647 360 Program: Generators
2F4(2)' 17 971 200 3 592 512 Program: Generators
25+8:(S3 × A6) 17 694 720 3 648 645 Program: Generators
31+6:23+4:32:2 5 038 848 12 812 800 Program: Generators
S10 3 628 800 17 791 488 Program: Generators
S10 3 628 800 17 791 488 Program: Generators
Program: Generators
M12 95 040 679 311 360 Program: Generators

### Maximal subgroups of Fi22:2

Subgroup Order Index Programs/reps
Fi22 64 561 751 654 400 2 Program: Generators
2.U6(2).2 36 787 322 880 3 510 Program: Generators
Program: Generators
O8+(2):S3 × 2 2 090 188 800 61 776 Program: Generators
210:M22:2 908 328 960 142 155 Program: Generators
27:S6(2) 185 794 560 694 980 Program: Generators
(2 × 21+8:U4(2):2):2 106 168 320 1 216 215 Program: Generators
U4(3).2.2 × S3 78 382 080 1 647 360 Program: Generators
2F4(2) 35 742 400 3 612 614 Program: Generators
25+8:(S3 × S6) 35 389 440 3 648 645 Program: Generators
35:(2 × U4(2):2) 25 194 240 5 125 120 Program: Generators
31+6:23+4:32.2.2 10 077 696 12 812 800 Program: Generators
G2(3):2 8 491 392 15 206 400 Program: Generators
M12:2 7 257 600 17 791 488 Program: Generators

## Conjugacy classes

### Conjugacy classes of Fi22

Conjugacy class Centraliser order Power up Class rep(s)
1A 64 561 751 654 400
2A 18 393 661 440 6A 6B 6E 10A 14A 18A 18B 18C 22A 22B 30A
2B 53 084 160 4A 4B 4C 4E 6C 6D 6I 8A 8B 8C 10B 12A 12B 12C 12D 12H 12I 12J 16A 16B 18D 20A 24A 24B
2C 1 769 472 4D 6F 6G 6H 6J 6K 8D 12E 12F 12G 12K
3A 19 595 520 6A 6D 6F 12B 12C 12D 12I 15A 21A 24A 24B 30A
3B 2 519 424 6B 6C 6G 9A 9B 12A 12E 12F 12H 18A 18B 18C 18D
3C 139 968 6E 6H 6I 6J 12G 12J
3D 17 496 6K 9C 12K
4A 110 592 8A 8B 12A 12B 12C 24A 24B
4B 46 080 12D 12J 20A
4C 12 288 8C 12H 16A 16B
4D 4 608 8D 12E 12F 12G 12K
4E 3 072 12I
5A 600 10A 10B 15A 20A 30A
6A 155 520 30A
6B 93 312 18A 18B 18C
6C 10 368 12A 12H 18D
6D 6 912 12B 12C 12D 12I 24A 24B
6E 3 888 Omitted owing to length.
6F 3 456 Omitted owing to length.
6G 3 456 12E 12F
6H 1 728 12G
6I 864 12J
6J 432 Omitted owing to length.
6K 216 12K
7A 42 14A 21A
8A 384 24B
8B 384 24A
8C 128 16A 16B
8D 32 abbabababbbabababbbabbb
9A 324 18A 18B 18D
9B 162 18C
9C 27 abbb
10A 60 30A
10B 40 20A
11A 22 11B2 22A 22B
11B 22 11A2 22A 22B
12A 864 Omitted owing to length.
12B 576 24A 24B
12C 576 ababbbabababbbabababbbabbbabbabababbbabbbabbbabababbbabbb
12D 288 ababbbabababbbabababbbabbbababbbbabababbb
12E 144 ababbbabbbabababbbabbbbabababbb
12F 144 abababbbabbbabababbbabababbbabbbbabababbbabbb
12G 144 babababbbabababbbabababbbabbbabbabababbbabbb
12H 96 babababbbbabababbbabababbbabababbbabbbabbabababbbabbb
12I 96 babababbbbabababbbabababbbabababbbabbbabbabababbbabbbab
12J 72 babababbbabababbbabababbbabbbabbabababbbabbbabbb
12K 36 abababbbabababbbabbbababbb
13A 13 13B2
13B 13 13A2
14A 14 babababbbabbb
15A 30 30A
16A 32 16B5
16B 32 16A5
18A 108 18B5
18B 108 18A5
18C 54 abbbabababbbabbb
18D 36 abbabababbbabbb
20A 20 abababbbabbb
21A 21 abb
22A 22 22B7
22B 22 22A7
24A 48 abababbbabababbbabbbabbabababbbabbb
24B 48 abababbb
30A 30 ababbb
13A-B aab
16A-B babababbb
18A-B ababababbb
22A-B abababbbabababbbabbb

### Conjugacy classes of Fi22:2

Conjugacy class Centraliser order Power up Class rep(s)
1A 129 123 503 308 800
2A 36 787 322 880 6A 6B 6E 10A 14A 18A 18B 22A 30A
2B 106 168 320 4A 4A 4B 4B 4C 4C 4E 4E 6C 6D 6I 8A 8B 8C 10B 12A 12B 12C 12D 12G 12H 12I 16A 18C 20A 24A 24B 4F 4G 4H 4I 8E 8F 8G 12K 12L 12M 12N 12O 12P 12Q 12R 12S 16B 20B 24C 24D 24E 24F 36A
2C 3 538 944 4D 4D 6F 6G 6H 6J 6K 8D 12E 12F 12J 4J 8H 12T 24G
3A 39 191 040 6A 6A 6D 6F 12B 12C 12D 12H 15A 21A 24A 24B 30A 6L 6M 6O 6Q 12K 12M 12O 12P 24C 24D 30B 30C 42A
3B 5 038 848 6B 6C 6G 9A 9B 12A 12E 12G 18A 18B 18C 6N 6R 12L 12Q 18D 18E 18F 24E 24F 36A
3C 279 936 6E 6H 6I 6J 12F 12I 6P 6U 6V 12N 12R 12S 24G
3D 34 992 6K 9C 12J 6S 6T 12T 18G
4A 221 184 8A 8B 12A 12B 12C 24A 24B 8E 8F 24C 24D
4B 92 160 12D 12I 20A
4C 24 576 8C 12G 16A 8G 16B 24E 24F
4D 9 216 8D 12E 12F 12J 8H 24G
4E 6 144 12H
5A 1 200 10A 10B 15A 20A 30A 10C 10D 10E 20B 30B 30C
6A 311 040 30A
6B 186 624 18A 18B
6C 20 736 12A 12G 18C 12L 12Q 24E 24F 36A
6D 13 824 12B 12C 12D 12H 24A 24B 12K 12M 12O 12P 24C 24D
6E 7 776
6F 6 912
6G 6 912 12E
6H 3 456 12F 24G
6I 1 728 12I 12N 12R 12S
6J 864
6K 432 12J 12T
7A 84 14A 21A 14B 14C 42A
8A 768 24B
8B 768 24A
8C 256 16A 16B
8D 64
9A 648 18A 18C 18D 36A
9B 324 18B 18E 18F
9C 54 18G
10A 120 30A
10B 80 20A 20B
11A 22 22A
12A 1 728
12B 1 152 24A 24B 24C 24D
12C 1 152
12D 576
12E 144
12F 288 24G
12G 192 24E 24F
12H 192
12I 144
12J 72
13A 13
14A 28
15A 60 30A 30B 30C
16A 32
18A 108
18B 108
18C 72 36A
20A 40
21A 42 42A
22A 22
24A 96
24B 96
30A 60
2D 2 090 188 800 6L 6M 6N 6P 6S 10C 14B 18D 18E 30B 42A
2E 6 635 520 6Q 6T 6U 10E 18G
2F 5 806 080 6O 6R 6V 10D 14C 18F 30C
4F 1 327 104 12K 12L 12M 12N 36A
4G 245 760 12O 20B
4H 18 432 12P 12Q 12R
4I 12 288 12S
4J 3 072 12T
6L 311 040 30B
6M 72 576 42A
6N 23 328 18D 18E
6O 8 640 30C
6P 7 776
6Q 6 912
6R 2 592 18F
6S 1 296
6T 1 296 18G
6U 864
6V 432
8E 768 24C
8F 768 24D
8G 768 24E 24F
8H 192 24G
10C 1 200 30B
10D 120 30C
10E 80
12K 6 912
12L 2 592 36A
12M 1 152
12N 864
12O 768
12P 576
12Q 288
12R 144
12S 96
12T 24
14B 84 42A
14C 28
16B 32
18D 108
18E 108
18F 36
18G 18
20B 40
24C 96
24D 96
24E 48 24F5
24F 48 24E5
24G 24
30B 60
30C 60
36A 36
42A 42