Conway group Co1


In the area of modern algebra known as group theory, the Conway group Co1 is a sporadic simple group of order

History and properties

Co1 is one of the 26 sporadic groups and was discovered by John Horton Conway in 1968. It is the largest of the three sporadic Conway groups and can be obtained as the quotient of Co0 by its center, which consists of the scalar matrices ±1. It also appears at the top of the automorphism group of the even 26-dimensional unimodular lattice II25,1. Some rather cryptic comments in Witt's collected works suggest that he found the Leech lattice and possibly the order of its automorphism group in unpublished work in 1940.
The outer automorphism group is trivial and the Schur multiplier has order 2.

Involutions

Co0 has 4 conjugacy classes of involutions; these collapse to 2 in Co1, but there are 4-elements in Co0 that correspond to a third class of involutions in Co1.
An image of a dodecad has a centralizer of type 211:M12:2, which is contained in a maximal subgroup of type 211:M24.
An image of an octad or 16-set has a centralizer of the form 21+8.O, a maximal subgroup.

Representations

The smallest faithful permutation representation of Co1 is on the 98280 pairs of norm 4 vectors.
The double cover Co0 has a 24 dimensional representation; when reduced modulo 2, this becomes a representation of the simple group Co1. The exterior square and symmetric tensor square of the 24 dimensional representation have dimension 276 and 299, respectively; in characteristic not 2, the former is the smallest faithful representation of the simple group Co1.
The centralizer of an involution of type 2B in the monster group is of the form 21+24 ·Co1. Under this subgroup, the 196883 dimensional representation of the monster reduces as follows: the 24x4096=98304 representation of 21+24 ·Co1, the 98280 dimensional representation of 224 ·Co1, and the 299 dimensional representation of Co1.
The Dynkin diagram of the even Lorentzian unimodular lattice II1,25 is isometric to the Leech lattice Λ, so the group of diagram automorphisms is split extension Λ,Co0 of affine isometries of the Leech lattice.

Maximal subgroups

found the 22 conjugacy classes of maximal subgroups of Co1, though there were some errors in this list, corrected by.
No.StructureOrderIndexComments
1Co242,305,421,312,000
= 218·36·53·7·11·23		98,280
= 23·33·5·7·13
23 · Suz:22,690,072,985,600
= 214·38·52·7·11·13		1,545,600
= 27·3·52·7·23		the lift to Aut = Co0 fixes a complex structure or changes it to the complex conjugate structure; also, top of Suzuki chain
3211:M24501,397,585,920
= 221·33·5·7·11·23		8,292,375
= 36·53·7·13		image of monomial subgroup from Aut, that subgroup stabilizing the standard frame of 48 vectors of form
4Co3495,766,656,000
= 210·37·53·7·11·23		8,386,560
= 211·32·5·7·13
521+8 · O89,181,388,800
= 221·35·52·7		46,621,575
= 34·52·7·11·13·23		centralizer of an involution of class 2A
6Fi21:S3 ≈ U6:S355,180,984,320
= 216·37·5·7·11		75,348,000
= 25·32·53·7·13·23		the lift to Aut is the symmetry group of a coplanar hexagon of 6 type 2 points
7:26,038,323,200
= 215·34·52·7·13		688,564,800
= 26·35·52·7·11·23		in Suzuki chain
822+12:1,981,808,640
= 221·33·5·7		2,097,970,875
= 36·53·7·11·13·23
924+12 · 849,346,560
= 221·34·5		4,895,265,375
= 35·53·72·11·13·23
1032 · U4.D8235,146,240
= 210·38·5·7		17,681,664,000
= 211·3·53·7·11·13·23
1136:2.M12138,568,320
= 27·39·5·11		30,005,248,000
= 214·53·72·13·23		holomorph of ternary Golay code
12:272,576,000
= 210·34·53·7		57,288,591,360
= 211·35·5·7·11·13·23		in Suzuki chain
1331+4:2.S4.225,194,240
= 28·39·5		165,028,864,000
= 213·53·72·11·13·23
14.24,354,560
= 29·35·5·7		954,809,856,000
= 212·34·53·7·11·13·23		in Suzuki chain
1533+4:2.2,519,424
= 27·39		1,650,288,640,000
= 214·54·72·11·13·23
16A9 × S31,088,640
= 27·35·5·7		3,819,239,424,000
= 214·34·53·7·11·13·23		in Suzuki chain
17:2846,720
= 27·33·5·72		4,910,450,688,000
= 214·36·53·11·13·23		in Suzuki chain
18.2144,000
= 27·32·53		28,873,450,045,440
= 214·37·5·72·11·13·23
1951+2:GL260,000
= 25·3·54		69,296,280,109,056
= 216·38·72·11·13·23
2053:.260,000
= 25·3·54		69,296,280,109,056
= 216·38·72·11·13·23
2172:3,528
= 23·32·72		1,178,508,165,120,000
= 218·37·54·11·13·23
2252:2A53,000
= 23·3·53		1,385,925,602,181,120
= 218·38·5·72·11·13·23