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

### 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

## Representations

### Representations of He

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
2058 Std Details
8330 Std Details
29155 Std Details
244800 Std Details
• Matrix representations
Char Ring Dimension ID Generators Description Link
0 Z 102 Std Details
Char Ring Dimension ID Generators Description Link
2 GF(2) 51 Std Details
2 GF(2) 101 Std Details
2 GF(2) 246 Std Details
2 GF(2) 680 Std Details
Char Ring Dimension ID Generators Description Link
3 GF(9) 51 Std Details
3 GF(9) 153 a Std Details
3 GF(9) 153 b Std Details
3 GF(3) 679 Std Details
Char Ring Dimension ID Generators Description Link
5 GF(25) 51 Std Details
5 GF(5) 104 Std Details
5 GF(25) 153 a Std Details
5 GF(25) 153 b Std Details
5 GF(5) 680 Std Details
5 GF(25) 925 a Std Details
5 GF(25) 925 b Std Details
Char Ring Dimension ID Generators Description Link
7 GF(7) 50 Std Details
7 GF(7) 153 Std Details
7 GF(7) 426 Std Details
7 GF(7) 798 Std Details
Char Ring Dimension ID Generators Description Link
17 GF(17) 102 Std Details
17 GF(17) 306 Std Details
17 GF(17) 680 Std Details

### Representations of He:2

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
2058 Std Details
8330 Std Details
• Matrix representations
Char Ring Dimension ID Generators Description Link
2 GF(2) 102 Std Details
2 GF(2) 202 Std Details
2 GF(2) 492 Std Details
2 GF(2) 680 Std Details
Char Ring Dimension ID Generators Description Link
3 GF(3) 102 Std Details
3 GF(3) 306 a Std Details
3 GF(3) 679 Std Details
Char Ring Dimension ID Generators Description Link
5 GF(5) 102 Std Details
5 GF(5) 104 a Std Details
5 GF(5) 306 a Std Details
5 GF(5) 680 Std Details
Char Ring Dimension ID Generators Description Link
7 GF(7) 50 Std Details
7 GF(7) 153 Std Details
7 GF(7) 426 Std Details
7 GF(7) 798 Std Details
Char Ring Dimension ID Generators Description Link
17 GF(17) 102 Std Details
17 GF(17) 306 a Std Details
17 GF(17) 680 Std Details

## Maximal subgroups

### Maximal subgroups of He

Subgroup Order Index Programs/reps
S4(4):2 1 958 400 2 058 Program: Standard generators
22.L3(4).S3 483 840 8 330 Program: Generators
Program: Generators
26:3.S6 138 240 29 155 Program: Generators
Program: Generators
26:3.S6 138 240 29 155 Program: Generators
Program: Generators
21+6.L3(2) 21 504 187 425 Program: Generators
72:2.L2(7) 16 464 244 800 Program: Generators
Program: Generators
3.S7 15 120 266 560 Program: Generators
71+2:(3 × S3) 6 174 652 800 Program: Generators
S4 × L3(2) 4 032 999 600 Program: Generators
7:3 × L3(2) 3 258 1 237 074 Program: Generators
Program: Generators
52:4A4 1 200 3 358 656 Program: 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)
1A 4 030 387 200
2A 161 280 4A 6A 10A 12A 14A 14B 28A 28B
2B 21 504 4B 4C 6B 8A 12B 14C 14D
3A 7 560 6A 12A 15A 21A 21B
3B 504 6B 12B 21C 21D
4A 672 12A 28A 28B
4B 384 12B
4C 128 8A
5A 300 10A 15A
6A 72 12A
6B 24 12B
7A 1 176 7B3 14A 14B 21C 21D 28A 28B
7B 1 176 7A3 14A 14B 21C 21D 28A 28B
7C 1 029 21A 21B
7D 98 7E3 14C 14D
7E 98 7D3 14C 14D
8A 16 abbb
10A 20 ababbabbbabb
12A 12 bababbabbbabbb
12B 12 abb
14A 56 14B3 28A 28B
14B 56 14A3 28A 28B
14C 14 14D3
14D 14 14C3
15A 15 ababb
17A 17 17B3
17B 17 17A3
21A 21 21B2
21B 21 21A2
21C 21 21D5
21D 21 21C5
28A 28 28B3
28B 28 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)
1A 8 060 774 400
2A 322 560 4A 6A 10A 12A 14A 28A 4D 12C 20A
2B 43 008 4B 4C 6B 8A 12B 14B 8B 8C 8D 16A 16B 24A 24B
3A 15 120 6A 12A 15A 21A 21B 6C 6D 12C 30A 42A 42B
3B 1 008 6B 12B 21C 6E 24A 24B
4A 1 344 12A 28A
4B 768 12B 8B 8C 8D 24A 24B
4C 256 8A 16A 16B
5A 600 10A 15A 10B 20A 30A
6A 144 12A 12C
6B 48 12B 24A 24B
7A 1 176 14A 21C 28A
7B 2 058 21A 21B 14C 42A 42B
7C 98 14B
8A 32 16A 16B
10A 40 20A
12A 24
12B 24 24A 24B
14A 56 28A
14B 14
15A 30 30A
17A 17
21A 42 21B2 42A 42B
21B 42 21A2 42A 42B
21C 21
28A 28
2C 30 240 6C 6D 6E 10B 14C 30A 42A 42B
4D 480 12C 20A
6C 15 120 30A 42A 42B
6D 144
6E 36
8B 192 8C3 24A 24B
8C 192 8B3 24A 24B
8D 32
10B 60 30A
12C 24
14C 42 42A 42B
16A 16 16B3
16B 16 16A3
20A 20
24A 24 24B5
24B 24 24A5
30A 30
42A 42 42B11
42B 42 42A11