# ATLAS: O'Nan group ON

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

### Standard generators

Standard generators of the O'Nan group ON 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 3ON are pre-images A and B where A has order 2, and B has order 4.
Standard generators of the automorphism group ON:2 are c and d where c is in class 2B, d is in class 4A, and cd has order 22.
Standard generators of 3ON: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

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.
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.

### Representations

The representations of ON available are
• a and b as 154 x 154 matrices over GF(3).
• a and b as 495 x 495 matrices over GF(3).
• a and b as 406 x 406 matrices over GF(7).
• a and b as 1618 x 1618 matrices over GF(7).
• a and b as 1869 x 1869 matrices over GF(31) - kindly provided by Markus Ottensmann.
• a and b as permutations on 122760 points.
The representations of 3ON available are
• A and B as 153 x 153 matrices over GF(4).
• A and B as 45 x 45 matrices over GF(7).
• A and B as 45 x 45 matrices over GF(7) - the dual of the above.
The representations of ON:2 available are
• c and d as 154 x 154 matrices over GF(3). In fact this group is only isoclinic to ON:2, and has structure ON:4, where an outer element of order 4 squares to -1.
• c and d as 154 x 154 matrices over GF(9).
The representations of 3ON:2 available are
• C and D as 306 x 306 matrices over GF(2).
• C and D as 90 x 90 matrices over GF(7).

### 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 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.