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

Specific
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