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

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
1140000StdDetails
• Matrix representations
Char Ring Dimension ID Generators Description Link
0Q(√5)133aStdDetails
0Q(√5)133bStdDetails
Char Ring Dimension ID Generators Description Link
2GF(4)132aStdDetails
2GF(4)132bStdDetails
2GF(4)133StdDetails
2GF(2)760StdDetails
2GF(4)2650aStdDetails
Char Ring Dimension ID Generators Description Link
3GF(9)133aStdDetails
3GF(9)133bStdDetails
3GF(3)760StdDetails
Char Ring Dimension ID Generators Description Link
5GF(5)133StdDetails
5GF(5)626StdDetails
5GF(5)627StdDetails
Char Ring Dimension ID Generators Description Link
7GF(49)133aStdDetails
7GF(49)133bStdDetails
7GF(7)760StdDetails
Char Ring Dimension ID Generators Description Link
11GF(11)133aStdDetails
11GF(11)133bStdDetails
11GF(11)760StdDetails
Char Ring Dimension ID Generators Description Link
19GF(19)133aStdDetails
19GF(19)133bStdDetails
19GF(19)760StdDetails

### Representations of HN:2

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
1140000StdDetails
• Matrix representations
Char Ring Dimension ID Generators Description Link
2GF(2)264StdDetails
5GF(5)133StdDetails

## Maximal subgroups

### Maximal subgroups of HN

Subgroup Order Index Programs/reps
A12 239 500 800 1 140 000Program: Generators
2.HS.2 177 408 000 1 539 000Program: Standard generators
U3(8):3 16 547 328 16 500 000Program: Generators
21+8.(A5 × A5).2 3 686 400 74 064 375Program: Generators
(D10 × U3(5)).2 2 520 000 108 345 600Program: Generators
51+4.21+4.5.4 2 000 000 136 515 456Program: Generators
26.U4(2) 1 658 880 164 587 500Program: 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 216Program: Generators
M12:2 190 080 1 436 400 000Program: Generators
M12:2 190 080 1 436 400 000Program: Generators
34:2.(A4 × A4).4 93 312 2 926 000 000
31+4:4.A5 58 320 4 681 600 000Program: 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)
1A273 030 912 000 000 (bababababbabababbababababababbabb)8
2A177 408 000 4B 6A 8A 10A 10B 10F 12A 14A 20C 22A 30A 40A 40B (abababababababb)6
2B3 686 400 4A 4C 6B 6C 8B 10C 10D 10E 10G 10H 12B 12C 20A 20B 20D 20E 30B 30C (bababababbabababbababababababbabb)4
3A544 320 6A 6B 12A 12B 15A 21A 30A (abababababababb)4
3B29 160 6C 9A 12C 15B 15C 30B 30C (ababababababbabb)4
4A15 360 8B 12B 20A 20B (bababababbabababbababababababbabb)2
4B5 760 8A 12A 20C 40A 40B (abababababababb)3
4C3 840 12C 20D 20E (ababababababbabb)3
5A630 000 10B 15A 20C 30A 35A 35B 40A 40B (abababbababbababababababbabb)8
5B500 000 10A 10C 20A 20B 25A 25B Omitted owing to length.
5C15 000 5D2 10D 10E 15B 15C 20D 20E 30B 30C (abababb)4
5D15 000 5C2 10D 10E 15B 15C 20D 20E 30B 30C (abababb)8
5E2 500 10F 10G 10H (abababbabababababb)2
6A1 440 12A 30A (abababababababb)2
6B576 12B Omitted owing to length.
6C360 12C 30B 30C (ababababababbabb)2
7A420 14A 21A 35A 35B (ababbababababababbabb)2
8A80 40A 40B (abababbababbababababababbabb)5
8B64 bababababbabababbababababababbabb
9A27 ababababbababababbabababbababababababbabb
10A2 000 abababababababbababababababbabbababbababababababbabbababababababbabb
10B1 200 20C 30A 40A 40B (abababbababbababababababbabb)4
10C800 20A 20B (ababababababb)2
10D600 10E3 20D 20E 30B 30C (abababb)6
10E600 10D3 20D 20E 30B 30C (abababb)2
10F100 abababababbababababababb
10G100 10H3 abababbabababababb
10H100 10G3 (abababbabababababb)3
11A22 22A (ab)2
12A72 abababababababb
12B48 abababbababababababbabbabababbbababababbabababbababababababbabb
12C12 ababababababbabb
14A28 ababbababababababbabb
15A180 30A (ababababbabababbababababababbabb)2
15B30 15C2 30B 30C (ababababbabb)4
15C30 15B2 30B 30C (ababababbabb)2
19A19 19B2 ababbababababababbabbababababababbabb
19B19 19A2 (ababbababababababbabbababababababbabb)2
20A80 20B3 ababababababb
20B80 20A3 (ababababababb)3
20C40 40A 40B (abababbababbababababababbabb)2
20D20 20E3 abababb
20E20 20D3 (abababb)3
21A21 abababbababababababbabb
22A22 ab
25A25 25B2 abababababb
25B25 25A2 (abababababb)2
30A60 ababababbabababbababababababbabb
30B30 30C7 ababababbabb
30C30 30B7 (ababababbabb)7
35A35 35B2 abababbababababababbabbabababb
35B35 35A2 (abababbababababababbabbabababb)2
40A40 40B3 abababbababbababababababbabb
40B40 40A3 (abababbababbababababababbabb)3

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### Conjugacy classes of HN:2

Conjugacy class Centraliser order Power up Class rep(s)
1A546 061 824 000 000
2A354 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
2B7 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
3A1 088 640 6A 6B 12A 12B 15A 21A 30A 6D 6E 12D 12E 24A 30C 42A 60A
3B58 320 6C 9A 12C 15B 30B 6F 18A 24B 24C
4A30 720 8B 12B 20A 20B 8C 8D 24A 40C 40D
4B11 520 8A 12A 20C 40A 8E 40B
4C7 680 12C 20D 8F 24B 24C
5A1 260 000 10B 15A 20C 30A 35A 40A 10G 20F 30C 40B 60A
5B1 000 000 10A 10C 20A 20B 25A 20E 20H 20I 40C 40D
5C15 000 10D 15B 20D 30B
5D5 000 10E 10F 10H 20G
6A2 880 12A 30A 12D 12E 60A
6B1 152 12B 24A
6C720 12C 30B 24B 24C
7A840 14A 21A 35A 14B 28A 42A
8A160 40A
8B128
9A54 18A
10A4 000 20E 20H
10B2 400 20C 30A 40A 20F 40B 60A
10C1 600 20A 20B 20I 40C 40D
10D600 20D 30B
10E200 20G
10F100
11A44 22A 44A 44B
12A144
12B96 24A
12C24 24B 24C
14A56 28A
15A360 30A 30C 60A
15B30 30B
19A19
20A160 20B3 40C 40D
20B160 20A3 40C 40D
20C80 40A 40B
20D20
21A42 42A
22A44 44A 44B
25A25
30A120 60A
30B30
35A35
40A40
2C7 257 600 6D 6E 6F 10G 10H 14B 18A 30C 42A
4D177 408 000 12D 20E 20F 20G 28A 44A 44B 60A
4E30 720 12E 20H
4F2 560 20I
6D30 240 30C 42A
6E864
6F324 18A
8C1 280 40C 40D
8D768 24A
8E160 40B
8F96 24B 24C
10G1 200 30C
10H100
12D1 440 60A
12E96
14B84 42A
18A18
20E2 000
20F1 200 60A
20G100
20H80
20I40
24A24
24B24 24C7
24C24 24B7
28A28
30C60
40B40
40C40 40D3
40D40 40C3
42A42
44A44 44B3
44B44 44A3
60A60

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