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 Download
O'N:2 Download

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 O'N

Representations of 3.O'N

Representations of O'N:2

Representations of 3.O'N:2

Representations of O'N:4

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)
1A 460 815 505 920
2A 161 280 4A 4B 6A 8A 8B 10A 12A 14A 16A 16B 16C 16D 20A 20B 28A 28B
3A 3 240 6A 12A 15A 15B
4A 80 640 12A 20A 20B 28A 28B
4B 256 8A 8B 16A 16B 16C 16D
5A 180 10A 15A 15B 20A 20B
6A 72 12A
7A 1 372 14A 28A 28B
7B 49 ababbbabbabababbababbbabababb
8A 32 16A 16B
8B 32 16C 16D
10A 20 20A 20B
11A 11 ab
12A 36 ababbb
14A 28 28A 28B
15A 45 15B2
15B 45 15A2
16A 16 16B3
16B 16 16A3
16C 16 16D3
16D 16 16C3
19A 19 19B4 19C2
19B 19 19A2 19C4
19C 19 19A4 19B2
20A 20 20B11
20B 20 20A11
28A 28 28B5
28B 28 28A5
31A 31 31B3
31B 31 31A3
15A-B ababbbabbabababbababbbabababbababbbabababb
16A-B abababbababbbabababb
16C-D abbabababbababbbabababb
19A-C abb
20A-B ababb
28A-B ababbbabababb
31A-B abababbabb

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Conjugacy classes of O'N:2

Conjugacy class Centraliser order Power up Class rep(s)
1A 921 631 011 840
2A 322 560 4A 4B 6A 8A 10A 12A 14A 16A 16B 20A 28A 28B 8B 8C 8D 24A 24B 56A 56B 56C 56D
3A 6 480 6A 12A 15A 15B 6B 24A 24B 30A 30B
4A 161 280 12A 20A 28A 28B 8B 8C 24A 24B 56A 56B 56C 56D
4B 512 8A 16A 16B 8D
5A 360 10A 15A 15B 20A 10B 10C 30A 30B
6A 144 12A 24A 24B
7A 2 744 14A 28A 28B 56A 56B 56C 56D
7B 98 14B
8A 32 16A 16B
10A 40 20A
11A 22 22A
12A 72 24A 24B
14A 56 28A 28B 56A 56B 56C 56D
15A 90 15B2 30A 30B
15B 90 15A2 30A 30B
16A 16 16B3
16B 16 16A3
19A 38 19B4 19C2 38A 38B 38C
19B 38 19A2 19C4 38A 38B 38C
19C 38 19A4 19B2 38A 38B 38C
20A 20
28A 56 28B5 56A 56B 56C 56D
28B 56 28A5 56A 56B 56C 56D
31A 31
2B 351 120 6B 10B 10C 14B 22A 30A 30B 38A 38B 38C
6B 60 30A 30B
8B 1 344 8C3 24A 24B 56A 56B 56C 56D
8C 1 344 8B3 24A 24B 56A 56B 56C 56D
8D 32
10B 60 10C3 30A 30B
10C 60 10B3 30A 30B
14B 14
22A 22
24A 24 24B5
24B 24 24A5
30A 30 30B7
30B 30 30A7
38A 38 38B9 38C3
38B 38 38A3 38C9
38C 38 38A9 38B3
56A 56 56B3 56C15 56D5
56B 56 56A3 56C5 56D15
56C 56 56A15 56B5 56D3
56D 56 56A5 56B15 56C3

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Miscellaneous Notes

Group Category Note
O'N:4 Technical note. This group is isoclinic to O'N:2, and has structure O'N:4, where an outer element of order 4 squares to −1