Order = 4030387200 = 210.33.52.73.17.
Mult = 1.
Out = 2.

## Porting notes

Porting incomplete.

## Standard generators

Standard generators of He are a, b where a is in class 2A, b is in class 7C and ab has order 17.

Standard generators of He:2 are c, d where c is in class 2B, d is in class 6C and cd has order 30.

## Black box algorithms

### Finding generators

Group Algorithm File
He
He:2

### Checking generators (semi-presentations)

Group Semi-presentation File
He 〈〈 a, b | o(a) = 2, o(b) = 7, o(ab) = 17, o(ababab2) = 23, o(z) = 10, o(az5) = 3; z = ab2abab2ab2 〉〉 Download
He:2 〈〈 c, d | o(c) = 2, o(d) = 6, o(cd) = 30, o(z) = 24, o(az12) = 17, o(t) = 15, o([t,y]) = 1; z = cddcddcd, y = b3, t = (u.ua)4((u.uabbab)2) 〉〉 Download

## Presentations

He a, b | a2 = b7 = (ab)17 = [a, b]6 = [a, b3]5 = [a, babab−1abab] = (ab)4ab2ab−3ababab−1ab3ab−2ab2 = 1 〉 Details

## Maximal subgroups

### Maximal subgroups of He

Subgroup Order Index Programs/reps
S4(4):2 1 958 400 2 058Program: Standard generators
22.L3(4).S3 483 840 8 330Program: Generators
Program: Generators
26:3.S6 138 240 29 155Program: Generators
Program: Generators
26:3.S6 138 240 29 155Program: Generators
Program: Generators
21+6.L3(2) 21 504 187 425Program: Generators
72:2.L2(7) 16 464 244 800Program: Generators
Program: Generators
3.S7 15 120 266 560Program: Generators
71+2:(3 × S3) 6 174 652 800Program: Generators
S4 × L3(2) 4 032 999 600Program: Generators
7:3 × L3(2) 3 258 1 237 074Program: Generators
Program: Generators
52:4A4 1 200 3 358 656Program: Generators

### Maximal subgroups of He:2

Subgroup Order Index Programs/reps
He Program: Standard generators
S4(4):4 Program: Standard generators
Program: Generators
22.L3(4).D12 Program: Generators mapping onto standard generators
Program: Generators
21+6.L3(2).2 Program: Generators mapping onto standard generators
72:2.L2(7).2 Program: Generators mapping onto standard generators
3.S7 × 2 Program: Generators mapping onto standard generators
(S5 × S5).2 Program: Generators
24+4.(S3 × S3).2 Program: Generators
71+2:(6 × S3) Program: Generators
S4 × L3(2):2 Program: Generators mapping onto standard generators
7:6 × L3(2) Program: Generators mapping onto standard generators
52:4S4 Program: Generators

## Conjugacy classes

### Conjugacy classes of He

Conjugacy class Centraliser order Power up Class rep(s)
1A4 030 387 200
2A161 280 4A 6A 10A 12A 14A 14B 28A 28B
2B21 504 4B 4C 6B 8A 12B 14C 14D
3A7 560 6A 12A 15A 21A 21B
3B504 6B 12B 21C 21D
4A672 12A 28A 28B
4B384 12B
4C128 8A
5A300 10A 15A
6A72 12A
6B24 12B
7A1 176 7B3 14A 14B 21C 21D 28A 28B
7B1 176 7A3 14A 14B 21C 21D 28A 28B
7C1 029 21A 21B
7D98 7E3 14C 14D
7E98 7D3 14C 14D
8A16 abbb
10A20 ababbabbbabb
12A12 bababbabbbabbb
12B12 abb
14A56 14B3 28A 28B
14B56 14A3 28A 28B
14C14 14D3
14D14 14C3
15A15 ababb
17A17 17B3
17B17 17A3
21A21 21B2
21B21 21A2
21C21 21D5
21D21 21C5
28A28 28B3
28B28 28A3
14C-D ababbabbbabbabb
17A-B ab
21A-B bababbabbbabbbabbb
21C-D ababbabbbabbb
28A-B ababbabbb

### Conjugacy classes of He:2

Conjugacy class Centraliser order Power up Class rep(s)
1A8 060 774 400
2A322 560 4A 6A 10A 12A 14A 28A 4D 12C 20A
2B43 008 4B 4C 6B 8A 12B 14B 8B 8C 8D 16A 16B 24A 24B
3A15 120 6A 12A 15A 21A 21B 6C 6D 12C 30A 42A 42B
3B1 008 6B 12B 21C 6E 24A 24B
4A1 344 12A 28A
4B768 12B 8B 8C 8D 24A 24B
4C256 8A 16A 16B
5A600 10A 15A 10B 20A 30A
6A144 12A 12C
6B48 12B 24A 24B
7A1 176 14A 21C 28A
7B2 058 21A 21B 14C 42A 42B
7C98 14B
8A32 16A 16B
10A40 20A
12A24
12B24 24A 24B
14A56 28A
14B14
15A30 30A
17A17
21A42 21B2 42A 42B
21B42 21A2 42A 42B
21C21
28A28
2C30 240 6C 6D 6E 10B 14C 30A 42A 42B
4D480 12C 20A
6C15 120 30A 42A 42B
6D144
6E36
8B192 8C3 24A 24B
8C192 8B3 24A 24B
8D32
10B60 30A
12C24
14C42 42A 42B
16A16 16B3
16B16 16A3
20A20
24A24 24B5
24B24 24A5
30A30
42A42 42B11
42B42 42A11