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
Fi22
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Fi22:2
Download

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

Group Presentation Link
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

Representations of 2.Fi22

Representations of 3.Fi22

Representations of Fi22:2

Representations of 2.Fi22:2

Representations of 3.Fi22:2

Representations of 6.Fi22

Maximal subgroups

Maximal subgroups of Fi22

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

Maximal subgroups of Fi22:2

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

Conjugacy classes

Conjugacy classes of Fi22

Conjugacy class Centraliser order Power up Class rep(s)
1A64 561 751 654 400
2A18 393 661 440 6A 6B 6E 10A 14A 18A 18B 18C 22A 22B 30A
2B53 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
2C1 769 472 4D 6F 6G 6H 6J 6K 8D 12E 12F 12G 12K
3A19 595 520 6A 6D 6F 12B 12C 12D 12I 15A 21A 24A 24B 30A
3B2 519 424 6B 6C 6G 9A 9B 12A 12E 12F 12H 18A 18B 18C 18D
3C139 968 6E 6H 6I 6J 12G 12J
3D17 496 6K 9C 12K
4A110 592 8A 8B 12A 12B 12C 24A 24B
4B46 080 12D 12J 20A
4C12 288 8C 12H 16A 16B
4D4 608 8D 12E 12F 12G 12K
4E3 072 12I
5A600 10A 10B 15A 20A 30A
6A155 520 30A
6B93 312 18A 18B 18C
6C10 368 12A 12H 18D
6D6 912 12B 12C 12D 12I 24A 24B
6E3 888 Omitted owing to length.
6F3 456 Omitted owing to length.
6G3 456 12E 12F
6H1 728 12G
6I864 12J
6J432 Omitted owing to length.
6K216 12K
7A42 14A 21A
8A384 24B
8B384 24A
8C128 16A 16B
8D32 abbabababbbabababbbabbb
9A324 18A 18B 18D
9B162 18C
9C27 abbb
10A60 30A
10B40 20A
11A22 11B2 22A 22B
11B22 11A2 22A 22B
12A864 Omitted owing to length.
12B576 24A 24B
12C576 ababbbabababbbabababbbabbbabbabababbbabbbabbbabababbbabbb
12D288 ababbbabababbbabababbbabbbababbbbabababbb
12E144 ababbbabbbabababbbabbbbabababbb
12F144 abababbbabbbabababbbabababbbabbbbabababbbabbb
12G144 babababbbabababbbabababbbabbbabbabababbbabbb
12H96 babababbbbabababbbabababbbabababbbabbbabbabababbbabbb
12I96 babababbbbabababbbabababbbabababbbabbbabbabababbbabbbab
12J72 babababbbabababbbabababbbabbbabbabababbbabbbabbb
12K36 abababbbabababbbabbbababbb
13A13 13B2
13B13 13A2
14A14 babababbbabbb
15A30 30A
16A32 16B5
16B32 16A5
18A108 18B5
18B108 18A5
18C54 abbbabababbbabbb
18D36 abbabababbbabbb
20A20 abababbbabbb
21A21 abb
22A22 22B7
22B22 22A7
24A48 abababbbabababbbabbbabbabababbbabbb
24B48 abababbb
30A30 ababbb
13A-B aab
16A-B babababbb
18A-B ababababbb
22A-B abababbbabababbbabbb

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Conjugacy classes of Fi22:2

Conjugacy class Centraliser order Power up Class rep(s)
1A129 123 503 308 800
2A36 787 322 880 6A 6B 6E 10A 14A 18A 18B 22A 30A
2B106 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
2C3 538 944 4D 4D 6F 6G 6H 6J 6K 8D 12E 12F 12J 4J 8H 12T 24G
3A39 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
3B5 038 848 6B 6C 6G 9A 9B 12A 12E 12G 18A 18B 18C 6N 6R 12L 12Q 18D 18E 18F 24E 24F 36A
3C279 936 6E 6H 6I 6J 12F 12I 6P 6U 6V 12N 12R 12S 24G
3D34 992 6K 9C 12J 6S 6T 12T 18G
4A221 184 8A 8B 12A 12B 12C 24A 24B 8E 8F 24C 24D
4B92 160 12D 12I 20A
4C24 576 8C 12G 16A 8G 16B 24E 24F
4D9 216 8D 12E 12F 12J 8H 24G
4E6 144 12H
5A1 200 10A 10B 15A 20A 30A 10C 10D 10E 20B 30B 30C
6A311 040 30A
6B186 624 18A 18B
6C20 736 12A 12G 18C 12L 12Q 24E 24F 36A
6D13 824 12B 12C 12D 12H 24A 24B 12K 12M 12O 12P 24C 24D
6E7 776
6F6 912
6G6 912 12E
6H3 456 12F 24G
6I1 728 12I 12N 12R 12S
6J864
6K432 12J 12T
7A84 14A 21A 14B 14C 42A
8A768 24B
8B768 24A
8C256 16A 16B
8D64
9A648 18A 18C 18D 36A
9B324 18B 18E 18F
9C54 18G
10A120 30A
10B80 20A 20B
11A22 22A
12A1 728
12B1 152 24A 24B 24C 24D
12C1 152
12D576
12E144
12F288 24G
12G192 24E 24F
12H192
12I144
12J72
13A13
14A28
15A60 30A 30B 30C
16A32
18A108
18B108
18C72 36A
20A40
21A42 42A
22A22
24A96
24B96
30A60
2D2 090 188 800 6L 6M 6N 6P 6S 10C 14B 18D 18E 30B 42A
2E6 635 520 6Q 6T 6U 10E 18G
2F5 806 080 6O 6R 6V 10D 14C 18F 30C
4F1 327 104 12K 12L 12M 12N 36A
4G245 760 12O 20B
4H18 432 12P 12Q 12R
4I12 288 12S
4J3 072 12T
6L311 040 30B
6M72 576 42A
6N23 328 18D 18E
6O8 640 30C
6P7 776
6Q6 912
6R2 592 18F
6S1 296
6T1 296 18G
6U864
6V432
8E768 24C
8F768 24D
8G768 24E 24F
8H192 24G
10C1 200 30B
10D120 30C
10E80
12K6 912
12L2 592 36A
12M1 152
12N864
12O768
12P576
12Q288
12R144
12S96
12T24
14B84 42A
14C28
16B32
18D108
18E108
18F36
18G18
20B40
24C96
24D96
24E48 24F5
24F48 24E5
24G24
30B60
30C60
36A36
42A42

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