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
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 122760 a Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 3 GF(3) 154 Std Details 3 GF(9) 342 a Std Details 3 GF(9) 342 b Std Details 3 GF(3) 495 Std Details 3 GF(3) 684 Std Details 7 GF(7) 406 Std Details 7 GF(7) 1618 Std Details 31 GF(31) 1869 Std Details
Representations of 3.O'N
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 368280 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(4) 153 Std Details 7 GF(7) 45 a Std Details 7 GF(7) 45 b Std Details
Representations of O'N:2
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 245520 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 3 GF(9) 154 Std Details 3 GF(3) 684 Std Details 3 GF(3) 990 Std Details 7 GF(7) 406 Std Details 7 GF(7) 1618 Std Details 31 GF(31) 1869 Std Details
Representations of 3.O'N:2
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 736560 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 306 Std Details 7 GF(7) 90 Std Details
Representations of O'N:4
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 3 GF(3) 154 Gens Details
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 |
Download words for class representatives.
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 |
Download words (if any exist) for class representatives.
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 |