Order = 273030912000000 = 214.36.56.7.11.19.
Mult = 1.
Out = 2.

Porting notes

Porting incomplete.

Standard generators

Standard generators of HN are a, b where a is in class 2A, b is in class 3B, ab has order 22 and ababb has order 5.

Standard generators of HN:2 are c, d where c is in class 2C, d is in class 5A and cd has order 42.

Black box algorithms

Finding generators

Group Algorithm File
HN Download
HN:2 Download

Checking generators (semi-presentations)

Group Semi-presentation File
HN 〈〈 a, b | o(a) = 2, o(b) = 3, o(ab) = 22, o(abab2) = 5, o(a((ab)11)abbababababb) = 5 〉〉 Download
HN:2 〈〈 c, d | o(c) = 2, o(d) = 5, o(cd) = 42, o(z) = 60, o(td) = 22, o(td2(td)3) = 22; z = cd3(cd)4, t = z30 〉〉 Download

Representations

Representations of HN

Representations of HN:2

Maximal subgroups

Maximal subgroups of HN

Subgroup Order Index Programs/reps
A12 239 500 800 1 140 000 Program: Generators
2.HS.2 177 408 000 1 539 000 Program: Standard generators
U3(8):3 16 547 328 16 500 000 Program: Generators
21+8.(A5 × A5).2 3 686 400 74 064 375 Program: Generators
(D10 × U3(5)).2 2 520 000 108 345 600 Program: Generators
51+4.21+4.5.4 2 000 000 136 515 456 Program: Generators
26.U4(2) 1 658 880 164 587 500 Program: Generators
(A6 × A6).D8 1 036 800 263 340 000
23+2+6.(3 × L3(2)) 1 032 192 264 515 625
52+1+2.4.A5 750 000 364 041 216 Program: Generators
M12:2 190 080 1 436 400 000 Program: Generators
M12:2 190 080 1 436 400 000 Program: Generators
34:2.(A4 × A4).4 93 312 2 926 000 000
31+4:4.A5 58 320 4 681 600 000 Program: Generators

Maximal subgroups of HN:2

Subgroup Order Index Programs/reps
HN Program: Standard generators
S12 Program: Generators
4.HS.2 Program: Generators
U3(8):6 Program: Generators
21+8.(A5 × A5).2.2 Program: Generators
5:4 × U3(5) Program: Generators
51+4.21+4.5.4.2 Program: Generators
26.U4(2).2 Program: Generators
(S6 × S6).2.2 Program: Generators
23+2+6.(3 × L3(2)).2 Program: Generators
52+1+2.4.A5.2 Program: Generators
34:2.(S4 × S4).2 Program: Generators
31+4:4.S5 Program: Generators

Conjugacy classes

Conjugacy classes of HN

Conjugacy class Centraliser order Power up Class rep(s)
1A 273 030 912 000 000 (bababababbabababbababababababbabb)8
2A 177 408 000 4B 6A 8A 10A 10B 10F 12A 14A 20C 22A 30A 40A 40B (abababababababb)6
2B 3 686 400 4A 4C 6B 6C 8B 10C 10D 10E 10G 10H 12B 12C 20A 20B 20D 20E 30B 30C (bababababbabababbababababababbabb)4
3A 544 320 6A 6B 12A 12B 15A 21A 30A (abababababababb)4
3B 29 160 6C 9A 12C 15B 15C 30B 30C (ababababababbabb)4
4A 15 360 8B 12B 20A 20B (bababababbabababbababababababbabb)2
4B 5 760 8A 12A 20C 40A 40B (abababababababb)3
4C 3 840 12C 20D 20E (ababababababbabb)3
5A 630 000 10B 15A 20C 30A 35A 35B 40A 40B (abababbababbababababababbabb)8
5B 500 000 10A 10C 20A 20B 25A 25B Omitted owing to length.
5C 15 000 5D2 10D 10E 15B 15C 20D 20E 30B 30C (abababb)4
5D 15 000 5C2 10D 10E 15B 15C 20D 20E 30B 30C (abababb)8
5E 2 500 10F 10G 10H (abababbabababababb)2
6A 1 440 12A 30A (abababababababb)2
6B 576 12B Omitted owing to length.
6C 360 12C 30B 30C (ababababababbabb)2
7A 420 14A 21A 35A 35B (ababbababababababbabb)2
8A 80 40A 40B (abababbababbababababababbabb)5
8B 64 bababababbabababbababababababbabb
9A 27 ababababbababababbabababbababababababbabb
10A 2 000 abababababababbababababababbabbababbababababababbabbababababababbabb
10B 1 200 20C 30A 40A 40B (abababbababbababababababbabb)4
10C 800 20A 20B (ababababababb)2
10D 600 10E3 20D 20E 30B 30C (abababb)6
10E 600 10D3 20D 20E 30B 30C (abababb)2
10F 100 abababababbababababababb
10G 100 10H3 abababbabababababb
10H 100 10G3 (abababbabababababb)3
11A 22 22A (ab)2
12A 72 abababababababb
12B 48 abababbababababababbabbabababbbababababbabababbababababababbabb
12C 12 ababababababbabb
14A 28 ababbababababababbabb
15A 180 30A (ababababbabababbababababababbabb)2
15B 30 15C2 30B 30C (ababababbabb)4
15C 30 15B2 30B 30C (ababababbabb)2
19A 19 19B2 ababbababababababbabbababababababbabb
19B 19 19A2 (ababbababababababbabbababababababbabb)2
20A 80 20B3 ababababababb
20B 80 20A3 (ababababababb)3
20C 40 40A 40B (abababbababbababababababbabb)2
20D 20 20E3 abababb
20E 20 20D3 (abababb)3
21A 21 abababbababababababbabb
22A 22 ab
25A 25 25B2 abababababb
25B 25 25A2 (abababababb)2
30A 60 ababababbabababbababababababbabb
30B 30 30C7 ababababbabb
30C 30 30B7 (ababababbabb)7
35A 35 35B2 abababbababababababbabbabababb
35B 35 35A2 (abababbababababababbabbabababb)2
40A 40 40B3 abababbababbababababababbabb
40B 40 40A3 (abababbababbababababababbabb)3

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Conjugacy classes of HN:2

Conjugacy class Centraliser order Power up Class rep(s)
1A 546 061 824 000 000
2A 354 816 000 4B 6A 8A 10A 10B 10E 12A 14A 20C 22A 30A 40A 4D 4E 8E 12D 12E 20E 20F 20G 20H 28A 40B 44A 44B 60A
2B 7 372 800 4A 4C 6B 6C 8B 10C 10D 10F 12B 12C 20A 20B 20D 30B 4F 8C 8D 8F 20I 24A 24B 24C 40C 40D
3A 1 088 640 6A 6B 12A 12B 15A 21A 30A 6D 6E 12D 12E 24A 30C 42A 60A
3B 58 320 6C 9A 12C 15B 30B 6F 18A 24B 24C
4A 30 720 8B 12B 20A 20B 8C 8D 24A 40C 40D
4B 11 520 8A 12A 20C 40A 8E 40B
4C 7 680 12C 20D 8F 24B 24C
5A 1 260 000 10B 15A 20C 30A 35A 40A 10G 20F 30C 40B 60A
5B 1 000 000 10A 10C 20A 20B 25A 20E 20H 20I 40C 40D
5C 15 000 10D 15B 20D 30B
5D 5 000 10E 10F 10H 20G
6A 2 880 12A 30A 12D 12E 60A
6B 1 152 12B 24A
6C 720 12C 30B 24B 24C
7A 840 14A 21A 35A 14B 28A 42A
8A 160 40A
8B 128
9A 54 18A
10A 4 000 20E 20H
10B 2 400 20C 30A 40A 20F 40B 60A
10C 1 600 20A 20B 20I 40C 40D
10D 600 20D 30B
10E 200 20G
10F 100
11A 44 22A 44A 44B
12A 144
12B 96 24A
12C 24 24B 24C
14A 56 28A
15A 360 30A 30C 60A
15B 30 30B
19A 19
20A 160 20B3 40C 40D
20B 160 20A3 40C 40D
20C 80 40A 40B
20D 20
21A 42 42A
22A 44 44A 44B
25A 25
30A 120 60A
30B 30
35A 35
40A 40
2C 7 257 600 6D 6E 6F 10G 10H 14B 18A 30C 42A
4D 177 408 000 12D 20E 20F 20G 28A 44A 44B 60A
4E 30 720 12E 20H
4F 2 560 20I
6D 30 240 30C 42A
6E 864
6F 324 18A
8C 1 280 40C 40D
8D 768 24A
8E 160 40B
8F 96 24B 24C
10G 1 200 30C
10H 100
12D 1 440 60A
12E 96
14B 84 42A
18A 18
20E 2 000
20F 1 200 60A
20G 100
20H 80
20I 40
24A 24
24B 24 24C7
24C 24 24B7
28A 28
30C 60
40B 40
40C 40 40D3
40D 40 40C3
42A 42
44A 44 44B3
44B 44 44A3
60A 60

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