ATLAS: Conway group Co1
Order = 4157776806543360000.
Mult = 2.
Out = 1.
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 2Co1 are
preimages A
and B where
B has order 3,
and ABABABABBABABBABB has order 33.
Black box algorithms
To find standard generators for Co1:
- Find any element of order 26 or 42. It powers up to a 2B-element x.
- Find any element of order divisible by 9
(i.e. 9, 18 or 36). It powers up to a 3C-element y.
- Find a conjugate a of x and a conjugate b of y, whose product has order 40,
such that ababb has order 6.
Representations
The representations of Co1 available are
- a and
b as
24 x 24 matrices over GF(2).
- a and
b as
276 x 276 matrices over GF(3).
- a and
b as
276 x 276 matrices over GF(5).
- a and
b as
276 x 276 matrices over GF(7).
- a and
b as
276 x 276 matrices over GF(11).
- a and
b as
276 x 276 matrices over GF(13).
- a and
b as
276 x 276 matrices over GF(23).
The representations of 2Co1 available are
- A and
B as
24 x 24 matrices over GF(3).
- A and
B as
24 x 24 matrices over GF(5).
- A and
B as
24 x 24 matrices over GF(7).
- A and
B as
24 x 24 matrices over GF(11).
- A and
B as
24 x 24 matrices over GF(13).
- A and
B as
24 x 24 matrices over GF(23).
- A and
B as
24 x 24 matrices over Z. (Also available in GAP format:
A and
B .)
Maximal subgroups
The maximal subgroups of Co1 are
- Co2, with standard generators
(abb)^22(ab)^20(ab)^18,
(abababbababb)^28(abababbabababababbababb)^3(abababbababb)^14.
- 3.Suz:2, with standard generators
(ab)^-2bab, (abb)^-2(abababbabababbab)^8abbabb.
- 2^11:M24, with generators
(ab)^-6(ababb)^3(ab)^6,
(abb)^-5(abababbabababbab)^4(abb)^5.
- Co3, with generators
(ab)^20,
(abb)^-3(abababbabb)^5.
- 2^1+8.O8+(2), with generators
(abbab)^3,
(abb)^-7b(abb)^7.
- U6(2):S3, with generators
abbab, b.
- (A4 x G2(4)):2
- 2^2+12:(A8 x S3)
- 2^4+12.(S3 x 3.S6)
- 3^2.U4(3).D8
- 3^6:2.M12
- (A5 x J2):2
- 3^1+4:2.S4(3).2
- (A6 x U3(3)).2
- 3^3+4:2.(S4 x S4)
- A9 x S3
- (A7 x L2(7)):2
- (D10 x (A5 x A5).2).2
- 5^1+2:GL2(5)
- 5^3:(4 x A5).2
- 5^2:2A5
- 7^2:(3 x 2S4)
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 23.03.99
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk