ATLAS: Conway group Co₁

Order = 4157776806543360000 = 2²¹.3⁹.5⁴.7².11.13.23.
Mult = 2.
Out = 1.

The following information is available for Co₁:

Standard generators.
Black box algorithms.
Representations.
Maximal subgroups.
Conjugacy classes.

Standard generators

Standard generators of the Conway group Co₁ 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.Co₁ = Co₀ are preimages A and B where B has order 3 and ABABABABBABABBABB has order 33.

Black box algorithms

Finding generators

To find standard generators for Co₁:

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 Co₁.

This algorithm is available in computer readable format: finder for Co₁.

Checking generators

To check that elements x and y of Co₁ 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^yyz²¹) = 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 Co₁.

Representations

The representations of Co₁ 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.Co₁ 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).

Maximal subgroups

The maximal subgroups of Co₁ are as follows.

Co₂, 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.
2¹¹:M₂₄, with generators (ab)^-6(ababb)^3(ab)^6, (abb)^-5(abababbabababbab)^4(abb)^5.
Co₃, with generators (ab)^20, (abb)^-3(abababbabb)^5(abb)^3.
2¹⁺⁸.O8+(2), with generators (ababb)^3, abb(abbababb)^2ababababbab.
U₆(2):S₃, with generators (abbab)^3, b.
(A₄ × G₂(4)):2.
2²⁺¹²:(A₈ × S₃).
2⁴⁺¹².(S₃ × 3.S₆).
3².U₄(3).D₈.
3⁶:2.M₁₂.
(A₅ × J₂):2.
3¹⁺⁴:2.S₄(3).2.
(A₆ × U₃(3)).2.
3³⁺⁴:2.(S₄ × S₄).
A₉ × S₃.
(A₇ × L₂(7)):2.
(D₁₀ × (A₅ × A₅).2).2.
5¹⁺²:GL₂(5).
5³:(4 × A₅).2.
7²:(3 × 2.S₄).
5²:2A₅.

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.

Go to main ATLAS (version 2.0) page.

Go to sporadic groups page.

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.

ATLAS: Conway group Co1