# ATLAS: O'Nan group O'N

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

The following information is available for O'N:

### Standard generators

Standard generators of the O'Nan group O'N are a and b where a has order 2, b is in class 4A, and ab has order 11.
Standard generators of the triple cover 3O'N are pre-images A and B where A has order 2, and B has order 4.
Standard generators of the automorphism group O'N:2 are c and d where c is in class 2B, d is in class 4A, and cd has order 22.
Standard generators of 3O'N:2 are preimages C and D, where D has order 4.
A pair of generators conjugate to a, b can be obtained as
a' = (cdd)^{-2}dd(cdd)^2, b' = d.

The outer automorphism of O'N may be realised by mapping (a,b) to (a,b-1).

### Black box algorithms

#### Finding generators

To find standard generators for O'N:

• Find any element of order 20 or 28. It powers up to a 2A-element x and a 4A-element y.
• Find a conjugate a of x and a conjugate b of y, whose product has order 11.
This algorithm is available in computer readable format: finder for O'N.

To find standard generators for O'N.2:

• Find any element of order 22, 30 or 38. It powers up to a 2B-element.
• Find any element of order 20, 28 or 56. This powers up to a 4A-element, y, say.
• Find a conjugate a of x and a conjugate b of y, whose product has order 22.
This algorithm is available in computer readable format: finder for O'N.2.

#### Checking generators

To check that elements x and y of O'N are standard generators:

• Check o(x) = 2
• Check o(y) = 4
• Check o(xy) = 11
• Let z = xyxy(yy(yy)xyxy)5
• Check o(z) = 5
• Check o([y,z]) = 1
This algorithm is available in computer readable format: checker for O'N.

To check that elements x and y of O'N.2 are standard generators:

• Check o(x) = 2
• Check o(y) = 4
• Check o(xy) = 22
• Let z = (xyyxyx(yy(yy)xyyxyx)7)2
• Check o([y,z]) = 1
This algorithm is available in computer readable format: checker for O'N.2.

### Representations

The representations of O'N available are
• Dimension 154 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 342 over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). Kindly provided by Jürgen Müller.
• Dimension 342 over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). Kindly provided by Jürgen Müller.
• Dimension 684 over GF(3) - reducible over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). Kindly provided by Jürgen Müller.
• Dimension 495 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 406 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 1618 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 1869 over GF(31): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 122760 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
The representations of 3O'N available are
The representations of O'N:2 available are
• The representations of O'N:4 available are as follows. This group is isoclinic to O'N:2, and has structure O'N:4, where an outer element of order 4 squares to -1.
The representations of 3O'N:2 available are

### Maximal subgroups

The maximal subgroups of O'N are
The maximal subgroups of O'N:2 are

### Conjugacy classes

A set of generators for the maximal cyclic subgroups of O'N can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.
A set of generators for the maximal cyclic subgroups of O'N:2 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.
Problems of algebraic conjugacy are not yet dealt with. Go to main ATLAS (version 2.0) page. Go to sporadic groups page. Go to old O'N page - ATLAS version 1. Anonymous ftp access is also available.

Version 2.0 created on 7th June 2000.
Last updated 7.1.05 by SJN.
Information checked to Level 0 on 07.06.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.