ATLAS: Conway group Co2
Order = 42305421312000.
Mult = 1.
Out = 1.
Standard generators
Standard generators of the Conway group Co2 are a
and b where
a is in class 2A, b is in class 5A,
and ab has order 28.
Black box algorithms
To find standard generators for Co2:
- Find any element of order 16 or 28. It powers up to a 2A-element x.
- Find any element of order 5, y, say.
- Find a conjugate a of x and a conjugate b of y, whose product has order 28.
- If abb has order 15, then y is in the wrong conjugacy class.
- Otherwise (abb has order 9), standard generators for Co2 have been obtained.
Representations
The representations available are
- a and
b as
24 x 24 matrices over GF(2) - indecomposable 1.22.1.
- a and
b as
22 x 22 matrices over GF(2).
- a and
b as
230 x 230 matrices over GF(2).
- a and
b as
748 x 748 matrices over GF(2).
- a and
b as
23 x 23 matrices over GF(3).
- a and
b as
23 x 23 matrices over GF(5).
- a and
b as
23 x 23 matrices over GF(7).
- a and
b as
23 x 23 matrices over GF(11).
- a and
b as
23 x 23 matrices over GF(23).
- a and
b as
permutations on 2300 points.
- a and
b as
permutations on 4600 points.
The maximal subgroups of Co2 are as follows. Words provided by Peter Walsh,
implemented and checked by Ibrahim Suleiman.
- U6(2):2, with generators
(ab)^7,
abababbababb)^-5(abababbab)^-1(ababb)^4abababbab(abababbababb)^5.
- 2^10:M22:2, with generators
here.
- McL, with generators
here.
- 2^1+8.S6(2), with generators
here.
- HS:2, with generators
here.
- 2^1+6+4:A8, with generators
here.
- U4(3).D8, with generators
here
- 2^4+10.(S5 x S3), with generators
here - this one doesn't work [U6(2):2 generated].
- M23, with standard generators
here.
- 3^1+4.2^1+4.S5, with generators
here.
- 5^1+2:4S4, with generators
here - this one doesn't work [Co2 generated].
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 21.12.99
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk