Conway group Co2

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 =

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:

  • Fi21:2 ≈ U6(2):2 - symmetry/reflection group of coplanar hexagon of 6 type 2 points. Fixes one hexagon in a rank 3 permutation representation of Co2 on 2300 such hexagons. Under this subgroup the hexagons are split into orbits of 1, 891, and 1408. Fi21 fixes a 2-2-2 triangle defining the plane.
  • 210:M22:2 has monomial representation described above; 210:M22 fixes a 2-2-4 triangle.
  • McL fixes a 2-2-3 triangle.
  • 21+8:Sp6(2) - centralizer of involution class 2A (trace -8)
  • HS:2 fixes a 2-3-3 triangle or exchanges its type 3 vertices with sign change.
  • (24 × 21+6).A8
  • U4(3):D8
  • 24+10.(S5 × S3)
  • M23 fixes a 2-3-4 triangle.
  • 31+4.21+4.S5
  • 51+2:4S4

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.

ClassOrder of centralizerCentralizerSize of classTrace
1Aall Co2124
2A743,178,24021+8:Sp6(2)32·52·11·23-8
2B41,287,68021+4:24.A82·34·5211·238
2C1,474,560210.A6.2223·34·52·7·11·230
3A466,56031+421+4A5211·52·7·11·23-3
3B155,5203×U4(2).2211·3·52·7·11·236
4A3,096,5764.26.U3(3).224·33·53·11·238
4B122,880[210]S525·35·52·7·11·23-4
4C73,728[213.32]25·34·53·7·11·234
4D49,152[214.3]24·35·53·7·11·230
4E6,144[211.3]27·35·53·7·11·234
4F6,144[211.3]27·35·53·7·11·230
4G1,280[28.5]210·36·52·7·11·230
5A3,00051+22A4215·35·7·11·23-1
5B6005×S5215·35·5·7·11·234
6A5,7603.21+4A5211·34·52·7·11·235
6B5,184[26.34]212·32·53·7·11·231
6C4,3206×S6213·33·52·7·11·234
6D3,456[27.33]211·33·53·7·11·23-2
6E576[26.32]212·34·53·7·11·232
6F288[25.32]213·34·53·7·11·230
7A567×D8215·36·53·11·2333
8A768[28.3]210·35·53·7·11·230
8B768[28.3]210·35·53·7·11·23-2
8C512[29]29·36·53·7·11·234
8D512[29]29·36·53·7·11·230
8E256[28]210·36·53·7·11·232
8F64[26]212·36·53·7·11·232
9A549×S3217·33·53·7·11·233
10A1205×2.A4215·35·52·7·11·233
10B6010×S3216·35·52·7·11·232
10C405×D8215·36·52·7·11·230
11A1111218·36·53·7·232
12A864[25.33]213·33·53·7·11·23-1
12B288[25.32]213·34·53·7·11·231
12C288[25.32]213·34·53·7·11·232
12D288[25.32]213·34·53·7·11·23-2
12E96[25.3]213·35·53·7·11·233
12F96[25.3]213·35·53·7·11·232
12G48[24.3]214·35·53·7·11·231
12H48[24.3]214·35·53·7·11·230
14A565×D8215·36·53·11·23-1
14B2814×2216·36·53·11·231power equivalent
14C2814×2216·36·53·11·231
15A3030217·35·52·7·11·231
15B3030217·35·52·7·11·232power equivalent
15C3030217·35·52·7·11·232
16A3216×2213·36·53·7·11·232
16B3216×2213·36·53·7·11·230
18A1818217·34·53·7·11·231
20A2020216·36·52·7·11·231
20B2020216·36·52·7·11·230
23A2323218·36·53·7·111power equivalent
23B2323218·36·53·7·111
24A2424215·35·53·7·11·230
24B2424215·35·53·7·11·231
28A2828216·36·53·11·231
30A3030217·35·52·7·11·23-1
30B3030217·35·52·7·11·230
30C3030217·35·52·7·11·230

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
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