ATLAS: Conway group Co_{3}
Order = 495766656000 = 2^{10}.3^{7}.5^{3}.7.11.23.
Mult = 1.
Out = 1.
Standard generators
Standard generators of the Conway group Co_{3} are a and
b where a is in class 3A, b is in class 4A, and
ab has order 14.
Black box algorithms
To find standard generators for Co_{3}:
- Find any element of order 9, 18, 24 or 30. It powers up to a 3A-element x.
- Find any element of order 20. It powers up to a 4A-element y.
- Find a conjugate a of x and a conjugate b of y
such that ab has order 14.
Representations
The representations of Co_{3} available are:
- a and
b as
permutations on 276 points.
- a and
b as
permutations on 552 points - imprimitive.
- a and
b as
permutations on 11178 points.
- a and
b as
permutations on 37950 points.
- a and
b as
permutations on 48600 points.
- a and
b as
permutations on 128800 points. (Reconstructed on 14/9/99.)
- a and
b as
22 × 22 matrices over GF(2).
- a and
b as
22 × 22 matrices over GF(3).
- a and
b as
126 × 126 matrices over GF(3).
- a and
b as
126 × 126 matrices over GF(3) - the dual of the above.
[NB: The ordering of the representations of degree 126 over GF(3) has not been finalised.]
- a and
b as
23 × 23 matrices over GF(5).
- a and
b as
23 × 23 matrices over GF(7).
- a and
b as
23 × 23 matrices over GF(11).
- a and
b as
23 × 23 matrices over GF(23).
- a and b as
23 × 23 matrices over Z.
Maximal subgroups
The maximal subgroups of Co_{3} are as follows. Words provided by
Peter Walsh, implemented and checked by Ibrahim Suleiman.
- McL:2, with generators
here.
- HS, with generators
here.
- U4(3).2.2, with generators
here.
- M23, with generators
here.
- 3^5:(2 × M11), with generators
here.
- 2.S6(2), with generators
here.
- U3(5):S3, with generators
here.
- 3^1+4:4S6, with generators
here.
- 2^4.A8, with generators
here.
- L3(4).D12, with generators
here.
- 2 × M12, with generators
here.
- [2^10.3^3], with generators
here, or (a different copy, with shorter words)
here.
- S3 × L2(8):3, with generators
here.
- A4 × S5, with generators
here, or (a different copy, with shorter words)
here.
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.
Return to main ATLAS page.
Last updated 14th September 1999,
R.A.Wilson, R.A.Parker and J.N.Bray