ATLAS: Conway group Co_{1}
Order = 4157776806543360000 = 2^{21}.3^{9}.5^{4}.7^{2}.11.13.23.
Mult = 2.
Out = 1.
The following information is available for Co_{1}:
Standard generators of the Conway group Co_{1} are a and
b where a is in class 2B, b is in class 3C,
ab has order 40 and ababb has order 6.
Standard generators of the double cover 2.Co_{1} = Co_{0}
are preimages A and B where B has order 3 and
ABABABABBABABBABB has order 33.
Finding generators
To find standard generators for Co_{1}:

Find any element of order 26 or 42. It powers up to a 2Belement x.
[The probability of success at each attempt is 17 in 546 (about 1 in 32).]

Find any element of order divisible by 9 (i.e. 9, 18 or 36). It powers up
to a 3Celement y.
[The probability of success at each attempt is 2 in 27 (about 1 in 14).]

Find a conjugate a of x and a conjugate b of y
such that ab has order 40 and ababb has order 6.
[The probability of success at each attempt is 54 in 8855 (about 1 in 164).]

Now a and b are standard generators of Co_{1}.
This algorithm is available in computer readable format:
finder for Co_{1}.
Checking generators
To check that elements x and y of Co_{1}
are standard generators:
 Check o(x) = 2
 Check o(y) = 3
 Check o(xy) = 40
 Check o(xyxyy) = 6
 Let z = xy(xyxyy)^{2}
 Check o(z) = 42
 Check o(x^{yy}z^{21}) = 11
 Let u = (xy)^{20}
 Let v = y^{xyxyxyxyy}
 Check o(uv) = 36
 Check o(uvvuvuv) = 18
This algorithm is available in computer readable format:
checker for Co_{1}.
The representations of Co_{1} available are:

Permutations on 98280 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

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

Dimension 274 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 276 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 298 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 276 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 299 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 276 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 299 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 276 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 299 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 276 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 299 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 276 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 299 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 2.Co_{1} available are:

Permutations on 196560 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 24 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 24 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 24 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 24 over GF(11):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 24 over GF(13):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 24 over GF(23):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 24 over Z:
A and
B (Meataxe),
A and B (GAP).
The maximal subgroups of Co_{1} are as follows.

Co_{2}, with standard generators
(abb)^22(ab)^20(abb)^18,
(abababbababb)^28(abababbabababababbababb)^6(abababbababb)^14.

3.Suz:2, with standard generators
(ab)^2bab, (abb)^2(abababbabababbab)^8abbabb.

2^{11}:M_{24}, with generators
(ab)^6(ababb)^3(ab)^6,
(abb)^5(abababbabababbab)^4(abb)^5.

Co_{3}, with generators
(ab)^20,
(abb)^3(abababbabb)^5(abb)^3.

2^{1+8}.O8+(2), with generators
(ababb)^3, abb(abbababb)^2ababababbab._{ }

U_{6}(2):S_{3}, with generators
(abbab)^3, b.

(A_{4} × G_{2}(4)):2.

2^{2+12}:(A_{8} × S_{3}).

2^{4+12}.(S_{3} × 3.S_{6}).

3^{2}.U_{4}(3).D_{8}.

3^{6}:2.M_{12}.

(A_{5} × J_{2}):2.

3^{1+4}:2.S_{4}(3).2.

(A_{6} × U_{3}(3)).2.

3^{3+4}:2.(S_{4} × S_{4}).

A_{9} × S_{3}.

(A_{7} × L_{2}(7)):2.

(D_{10} × (A_{5} × A_{5}).2).2.

5^{1+2}:GL_{2}(5).

5^{3}:(4 × A_{5}).2.

7^{2}:(3 × 2.S_{4}).

5^{2}:2A_{5}.
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.
Go to main ATLAS (version 2.0) page.
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Go to old Co1 page  ATLAS version 1.
Anonymous ftp access is also available on
for.mat.bham.ac.uk, user atlasftp, password atlasftp.
Files can be found in directory v2.0 and subdirectories.
Version 2.0 created on 1st June 1999.
Last updated 6.1.05 by SJN.
Information checked to
Level 1 on 11.06.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.