ART

In the area of modern algebra known as group theory, the Conway group Co2 is a sporadic simple group of order

   218 · 36 · 53 · 7 · 11 · 23
= 42305421312000
≈ 4×1013.


History and properties

Co2 is one of the 26 sporadic groups and was discovered by (Conway 1968, 1969) as the group of automorphisms of the Leech lattice Λ fixing a lattice vector of type 2. It is thus a subgroup of Co0. It is isomorphic to a subgroup of Co1. The direct product 2×Co2 is maximal in Co0.

The Schur multiplier and the outer automorphism group are both trivial.

Representations

Co2 acts as a rank 3 permutation group on 2300 points. These points can be identified with planar hexagons in the Leech lattice having 6 type 2 vertices.

Co2 acts on the 23-dimensional even integral lattice with no roots of determinant 4, given as a sublattice of the Leech lattice orthogonal to a norm 4 vector. Over the field with 2 elements it has a 22-dimensional faithful representation; this is the smallest faithful representation over any field.

Feit (1974) showed that if a finite group has an absolutely irreducible faithful rational representation of dimension 23 and has no subgroups of index 23 or 24 then it is contained in either Z/2Z × Co2 or Z/2Z × Co3.

The Mathieu group M23 is isomorphic to a maximal subgroup of Co2 and one representation, in permutation matrices, fixes the type 2 vector u = (-3,123). A block sum ζ of the involution η =

\( {{\mathbf 1}/2}\left({\begin{matrix}1&-1&-1&-1\\-1&1&-1&-1\\-1&-1&1&-1\\-1&-1&-1&1\end{matrix}}\right) \)

and 5 copies of -η also fixes the same vector. Hence Co2 has a convenient matrix representation inside the standard representation of Co0. The trace of ζ is -8, while the involutions in M23 have trace 8.

A 24-dimensional block sum of η and -η is in Co0 if and only if the number of copies of η is odd.

Another representation fixes the vector v = (4,-4,022). A monomial and maximal subgroup includes a representation of M22:2, where any α interchanging the first 2 co-ordinates restores v by then negating the vector. Also included are diagonal involutions corresponding to octads (trace 8), 16-sets (trace -8), and dodecads (trace 0). It can be shown that Co2 has just 3 conjugacy classes of involutions. η leaves (4,-4,0,0) unchanged; the block sum ζ provides a non-monomial generator completing this representation of Co2.

There is an alternate way to construct the stabilizer of v. Now u and u+v = (1,-3,122) are vertices of a 2-2-2 triangle (vide infra). Then u, u+v, v, and their negatives form a coplanar hexagon fixed by ζ and M22; these generate a group Fi21 ≈ U6(2). α (vide supra) extends this to Fi21:2, which is maximal in Co2. Lastly, Co0 is transitive on type 2 points, so that a 23-cycle fixing u has a conjugate fixing v, and the generation is completed.

Maximal subgroups

Some maximal subgroups fix or reflect 2-dimensional sublattices of the Leech lattice. It is usual to define these planes by h-k-l triangles: triangles including the origin as a vertex, with edges (differences of vertices) being vectors of types h, k, and l.

Wilson (2009) found the 11 conjugacy classes of maximal subgroups of Co2 as follows:

Conjugacy classes

Traces of matrices in a standard 24-dimensional representation of Co2 are shown.[1] The names of conjugacy classes are taken from the Atlas of Finite Group Representations. [2]

Centralizers of unknown structure are indicated with brackets.

Class Order of centralizer Centralizer Size of class Trace
1A all Co2 1 24
2A 743,178,240 21+8:Sp6(2) 32·52·11·23 -8
2B 41,287,680 21+4:24.A8 2·34·5211·23 8
2C 1,474,560 210.A6.22 23·34·52·7·11·23 0
3A 466,560 31+421+4A5 211·52·7·11·23 -3
3B 155,520 3×U4(2).2 211·3·52·7·11·23 6
4A 3,096,576 4.26.U3(3).2 24·33·53·11·23 8
4B 122,880 [210]S5 25·35·52·7·11·23 -4
4C 73,728 [213.32] 25·34·53·7·11·23 4
4D 49,152 [214.3] 24·35·53·7·11·23 0
4E 6,144 [211.3] 27·35·53·7·11·23 4
4F 6,144 [211.3] 27·35·53·7·11·23 0
4G 1,280 [28.5] 210·36·52·7·11·23 0
5A 3,000 51+22A4 215·35·7·11·23 -1
5B 600 5×S5 215·35·5·7·11·23 4
6A 5,760 3.21+4A5 211·34·52·7·11·23 5
6B 5,184 [26.34] 212·32·53·7·11·23 1
6C 4,320 6×S6 213·33·52·7·11·23 4
6D 3,456 [27.33] 211·33·53·7·11·23 -2
6E 576 [26.32] 212·34·53·7·11·23 2
6F 288 [25.32] 213·34·53·7·11·23 0
7A 56 7×D8 215·36·53·11·233 3
8A 768 [28.3] 210·35·53·7·11·23 0
8B 768 [28.3] 210·35·53·7·11·23 -2
8C 512 [29] 29·36·53·7·11·23 4
8D 512 [29] 29·36·53·7·11·23 0
8E 256 [28] 210·36·53·7·11·23 2
8F 64 [26] 212·36·53·7·11·23 2
9A 54 9×S3 217·33·53·7·11·23 3
10A 120 5×2.A4 215·35·52·7·11·23 3
10B 60 10×S3 216·35·52·7·11·23 2
10C 40 5×D8 215·36·52·7·11·23 0
11A 11 11 218·36·53·7·23 2
12A 864 [25.33] 213·33·53·7·11·23 -1
12B 288 [25.32] 213·34·53·7·11·23 1
12C 288 [25.32] 213·34·53·7·11·23 2
12D 288 [25.32] 213·34·53·7·11·23 -2
12E 96 [25.3] 213·35·53·7·11·23 3
12F 96 [25.3] 213·35·53·7·11·23 2
12G 48 [24.3] 214·35·53·7·11·23 1
12H 48 [24.3] 214·35·53·7·11·23 0
14A 56 5×D8 215·36·53·11·23 -1
14B 28 14×2 216·36·53·11·23 1 power equivalent
14C 28 14×2 216·36·53·11·23 1
15A 30 30 217·35·52·7·11·23 1
15B 30 30 217·35·52·7·11·23 2 power equivalent
15C 30 30 217·35·52·7·11·23 2
16A 32 16×2 213·36·53·7·11·23 2
16B 32 16×2 213·36·53·7·11·23 0
18A 18 18 217·34·53·7·11·23 1
20A 20 20 216·36·52·7·11·23 1
20B 20 20 216·36·52·7·11·23 0
23A 23 23 218·36·53·7·11 1 power equivalent
23B 23 23 218·36·53·7·11 1
24A 24 24 215·35·53·7·11·23 0
24B 24 24 215·35·53·7·11·23 1
28A 28 28 216·36·53·11·23 1
30A 30 30 217·35·52·7·11·23 -1
30B 30 30 217·35·52·7·11·23 0
30C 30 30 217·35·52·7·11·23 0

References

Conway, John Horton (1968), "A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups", Proceedings of the National Academy of Sciences of the United States of America, 61 (2): 398–400, doi:10.1073/pnas.61.2.398, MR 0237634, PMC 225171, PMID 16591697
Conway, John Horton (1969), "A group of order 8,315,553,613,086,720,000", The Bulletin of the London Mathematical Society, 1: 79–88, doi:10.1112/blms/1.1.79, ISSN 0024-6093, MR 0248216
Conway, John Horton (1971), "Three lectures on exceptional groups", in Powell, M. B.; Higman, Graham (eds.), Finite simple groups, Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969., Boston, MA: Academic Press, pp. 215–247, ISBN 978-0-12-563850-0, MR 0338152 Reprinted in Conway & Sloane (1999, 267–298)
Conway, John Horton; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, 290 (3rd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-2016-7, ISBN 978-0-387-98585-5, MR 0920369
Feit, Walter (1974), "On integral representations of finite groups", Proceedings of the London Mathematical Society, Third Series, 29: 633–683, doi:10.1112/plms/s3-29.4.633, ISSN 0024-6115, MR 0374248
Thompson, Thomas M. (1983), From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs, 21, Mathematical Association of America, ISBN 978-0-88385-023-7, MR 0749038
Conway, John Horton; Parker, Richard A.; Norton, Simon P.; Curtis, R. T.; Wilson, Robert A. (1985), Atlas of finite groups, Oxford University Press, ISBN 978-0-19-853199-9, MR 0827219
Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-03516-0, ISBN 978-3-540-62778-4, MR 1707296
Wilson, Robert A. (1983), "The maximal subgroups of Conway's group ·2", Journal of Algebra, 84 (1): 107–114, doi:10.1016/0021-8693(83)90069-8, ISSN 0021-8693, MR 0716772
Wilson, Robert A. (2009), The finite simple groups., Graduate Texts in Mathematics 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 1203.20012

Specific

Wilson (1983)
http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co2/#ccls

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