Order = 460815505920 = 29.34.5.73.11.19.31.
Mult = 3.
Out = 2.

## Porting notes

Porting incomplete.

## Standard generators

Standard generators of O'N are a, b where a has order 2, b is in class 4A and ab has order 11.

Standard generators of 3.O'N are preimages A, B where A has order 2 and B has order 4.

Standard generators of O'N:2 are c, d where c is in class 2B, d is in class 4A and cd has order 22.

Standard generators of 3.O'N:2 are preimages C, D where D has order 4.

Generators of O'N:4 are c, d.

## Black box algorithms

### Finding generators

Group Algorithm File
O'N
O'N:2

### Checking generators (semi-presentations)

Group Semi-presentation File
O'N 〈〈 a, b | o(a) = 2, o(b) = 4, o(ab) = 11, o(z) = 5, o([b,z]) = 1; z = abab(b2(b2)abab)5 〉〉 Download
O'N:2 〈〈 c, d | o(c) = 2, o(d) = 4, o(cd) = 22, o([b,z]) = 1; z = (cddcdc(d2(d2)cddcdc)7)2 〉〉 Download

## Representations

### Representations of 3.O'N

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
368280StdDetails
• Matrix representations
Char Ring Dimension ID Generators Description Link
2GF(4)153StdDetails
7GF(7)45aStdDetails
7GF(7)45bStdDetails

### Representations of 3.O'N:2

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
736560StdDetails
• Matrix representations
Char Ring Dimension ID Generators Description Link
2GF(2)306StdDetails
7GF(7)90StdDetails

## Maximal subgroups

### Maximal subgroups of O'N

Subgroup Order Index Programs/reps
L3(7):2 3 753 792 122 760 Program: Standard generators
L3(7):2 3 753 792 122 760 Program: Standard generators
J1 175 560 2 624 832 Program: Standard generators
42.L3(4):21 161 280 2 857 239 Program: Generators
(32:4 × A6).2 25 920 17 778 376 Program: Generators
34:21+4.D10 25 920 17 778 376 Program: Generators
L2(31) 14 880 30 968 784 Program: Generators
L2(31) 14 880 30 968 784 Program: Generators
43.L3(2) 10 752 42 858 585 Program: Generators
M11 7 920 58 183 776 Program: Standard generators
M11 7 920 58 183 776 Program: Standard generators
A7 2 520 182 863 296 Program: Generators
A7 2 520 182 863 296 Program: Generators

### Maximal subgroups of O'N:2

Subgroup Order Index Programs/reps
O'N Program: Standard generators
71+2:(3 × D16) Program: Generators
J1 × 2 Program: Generators
4.L3(4).2.2 Program: Generators
(32:4 × A6).2.2 Program: Generators
34:21+4.D10.2 Program: Generators
31:30 Program: Generators
43.(L3(2) × 2) Program: Generators
L2(7):2 Program: Generators
A6:2 Program: Generators

## Conjugacy classes

### Conjugacy classes of O'N

Conjugacy class Centraliser order Power up Class rep(s)
1A460 815 505 920
2A161 280 4A 4B 6A 8A 8B 10A 12A 14A 16A 16B 16C 16D 20A 20B 28A 28B
3A3 240 6A 12A 15A 15B
4A80 640 12A 20A 20B 28A 28B
4B256 8A 8B 16A 16B 16C 16D
5A180 10A 15A 15B 20A 20B
6A72 12A
7A1 372 14A 28A 28B
7B49 ababbbabbabababbababbbabababb
8A32 16A 16B
8B32 16C 16D
10A20 20A 20B
11A11 ab
12A36 ababbb
14A28 28A 28B
15A45 15B2
15B45 15A2
16A16 16B3
16B16 16A3
16C16 16D3
16D16 16C3
19A19 19B4 19C2
19B19 19A2 19C4
19C19 19A4 19B2
20A20 20B11
20B20 20A11
28A28 28B5
28B28 28A5
31A31 31B3
31B31 31A3
15A-B ababbbabbabababbababbbabababbababbbabababb
16A-B abababbababbbabababb
16C-D abbabababbababbbabababb
19A-C abb
20A-B ababb
28A-B ababbbabababb
31A-B abababbabb

### Conjugacy classes of O'N:2

Conjugacy class Centraliser order Power up Class rep(s)
1A921 631 011 840
2A322 560 4A 4B 6A 8A 10A 12A 14A 16A 16B 20A 28A 28B 8B 8C 8D 24A 24B 56A 56B 56C 56D
3A6 480 6A 12A 15A 15B 6B 24A 24B 30A 30B
4A161 280 12A 20A 28A 28B 8B 8C 24A 24B 56A 56B 56C 56D
4B512 8A 16A 16B 8D
5A360 10A 15A 15B 20A 10B 10C 30A 30B
6A144 12A 24A 24B
7A2 744 14A 28A 28B 56A 56B 56C 56D
7B98 14B
8A32 16A 16B
10A40 20A
11A22 22A
12A72 24A 24B
14A56 28A 28B 56A 56B 56C 56D
15A90 15B2 30A 30B
15B90 15A2 30A 30B
16A16 16B3
16B16 16A3
19A38 19B4 19C2 38A 38B 38C
19B38 19A2 19C4 38A 38B 38C
19C38 19A4 19B2 38A 38B 38C
20A20
28A56 28B5 56A 56B 56C 56D
28B56 28A5 56A 56B 56C 56D
31A31
2B351 120 6B 10B 10C 14B 22A 30A 30B 38A 38B 38C
6B60 30A 30B
8B1 344 8C3 24A 24B 56A 56B 56C 56D
8C1 344 8B3 24A 24B 56A 56B 56C 56D
8D32
10B60 10C3 30A 30B
10C60 10B3 30A 30B
14B14
22A22
24A24 24B5
24B24 24A5
30A30 30B7
30B30 30A7
38A38 38B9 38C3
38B38 38A3 38C9
38C38 38A9 38B3
56A56 56B3 56C15 56D5
56B56 56A3 56C5 56D15
56C56 56A15 56B5 56D3
56D56 56A5 56B15 56C3