ATLAS: Conway group Co_{2}
Order = 42305421312000 = 2^{18}.3^{6}.5^{3}.7.11.23.
Mult = 1.
Out = 1.
The following information is available for Co_{2}:
Standard generators of the Conway group Co_{2} are a and
b where a is in class 2A, b is in class 5A and
ab has order 28.
Finding generators
To find standard generators for Co_{2}:

Find any element of order 16, 18 or 28. This powers up to a 2Aelement x.
[The probability of success at each attempt is 155 in 1008 (about 1 in 7).]

Find any element of order 15 or 30, This powers up to y of order 5.
[The probability of success at each attempt is 1 in 5, and the probability
that y ends up being in class 5A is 2 in 3.]

Find a conjugate a of x and a conjugate b of y, whose product has order 28.
[If y is in class 5A, then the probability of success at
each attempt is 40 in 759 (about 1 in 19).
If y is in class 5B, then the probability of success at
each attempt is 8 in 759 (about 1 in 95).]
If you have still not succeeded after (say) 35 attempts at this step, you
begin to suspect that y is in the wrong conjugacy class, so go
back to Step 2.

If abb has order 15, then y is in the wrong conjugacy class, so
go back to Step 2.

Otherwise abb has order 9 and standard generators for Co_{2} have been obtained.
This algorithm is available in computer readable format:
finder for Co_{2}.
Checking generators
To check that elements x and y of Co_{2}
are standard generators:
 Check o(x) = 2
 Check o(y) = 5
 Check o(xy) = 28
 Check o(x(xy)^{14}) = 3
This algorithm is available in computer readable format:
checker for Co_{2}.
A presentation for Co_{2} in terms of its standard generators is given below.
< a, b  a^{2} = b^{5} = (ab^{2})^{9} = [a, b]^{4} = [a, b^{2}]^{4} = [a, bab]^{3} = [a, bab^{2}ab]^{2} = [a, bab^{2}]^{3} = [a, b^{2}abab^{2}]^{2} = (abab^{2}ab^{1}ab^{2})^{7} = 1 >.
This presentation is available in Magma format as follows:
Co2 on a and b.
The representations of Co_{2} available are:

Permutations on 2300 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 4600 points  imprimitive:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 22 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 24 over GF(2)  reducible:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 230 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 748 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 748 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 23 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 253 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 275 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 23 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 253 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 275 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 23 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 253 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 275 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 23 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 253 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 275 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 23 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 253 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 274 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The maximal subgroups of Co_{2} are as follows. Words provided by
Peter Walsh, implemented and checked by Ibrahim Suleiman.

U_{6}(2):2, with generators
(ab)^7,
abababbababb)^5(abababbab)^1(ababb)^4abababbab(abababbababb)^5.
Er . . . try this one  it's shorter
(a, abab).

2^{10}:M_{22}:2, with generators
here.

McL, with generators
here.

2^{1+8}.S_{6}(2), with generators
here.

HS:2, with generators
here.

(2^{4} × 2^{1+6}).A_{8}, with generators
here.
The quotient of this group by its centre is a split extension 2^{10}:A_{8}, and the 2^{10} regared as a module for
the A_{8} is a uniserial module 4.6.
There is just one class of complementary A_{8} in this image,
and they become 2.A_{8} in (2^{4} × 2^{1+6}).A_{8}.

U_{4}(3):D_{8}, with generators
here

2^{4+10}.(S_{5} × S_{3}), with generators
[a, abbbababbb], abababbabbab.

M_{23}, with standard generators
here.

3^{1+4}.2^{1+4}.S_{5}, with generators
here.

5^{1+2}:4S_{4}, with generators
(abb)^3, (ab)^6(abbbb)^8(abababbbabbb)^5(abbbb)^8(ab)^6.
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: the linkage 15BC23AB is compatible with the
748dimensional representation mod 2. The choice of 14BC is made `without
loss of generality'.
Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to old Co2 page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 26th May 1999.
Last updated 6.1.05 by SJN.
Information checked to
Level 1 on 31.05.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.