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Conway group Co3

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Conway group Co3

In the area of modern algebra known as group theory, the Conway group Co3 is a sporadic simple group of order

Contents

   210 · 37 · 53 · 7 · 11 · 23 = 495766656000 ≈ 5×1011.

History and properties

Co3 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 3, thus length √ 6. It is thus a subgroup of Co0. It is isomorphic to a subgroup of Co1. The direct product 2xCo3 is maximal in Co0.

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

Representations

Co3 acts on the unique 23-dimensional even lattice of determinant 4 with no roots, given by the orthogonal complement of a norm 4 vector of the Leech lattice. This gives 23-dimensional representations over any field; over fields of characteristic 2 or 3 this can be reduced to a 22-dimensional faithful representation.

Co3 has a doubly transitive permutation representation on 276 points.

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.

Maximal subgroups

Finkelstein (1973) found the 14 conjugacy classes of maximal subgroups of Co3 as follows:

  • McL:2 – can transpose type 2 points of conserved 2-2-3 triangle. Co3 has a doubly transitive permutation representation on 276 type 2-2-3 triangles containing a fixed type 3 point.
  • HS – fixes 2-3-3 triangle.
  • U4(3).22
  • M23
  • 35:(2 × M11)
  • 2.Sp6(2) – centralizer of involution class 2A (trace 8), which moves 240 of the 276 type 2-2-3 triangles
  • U3(5):S3
  • 31+4:4S6
  • 24.A8
  • PSL(3,4):(2 × S3)
  • 2 × M12 – centralizer of involution class 2B (trace 0), which moves 264 of the 276 type 2-2-3 triangles
  • [210.33]
  • S3 × PSL(2,8):3
  • A4 × S5

    • Generalized Monstrous Moonshine

    In analogy to monstrous moonshine for the monster M, for Co3, the relevant McKay-Thompson series is T 4 A ( τ ) where one can set the constant term a(0) = 24 ( A097340),

    j 4 A ( τ ) = T 4 A ( τ ) + 24 = ( η 2 ( 2 τ ) η ( τ ) η ( 4 τ ) ) 24 = ( ( η ( τ ) η ( 4 τ ) ) 4 + 4 2 ( η ( 4 τ ) η ( τ ) ) 4 ) 2 = 1 q + 24 + 276 q + 2048 q 2 + 11202 q 3 + 49152 q 4 + …

    and η(τ) is the Dedekind eta function.

    References

    Conway group Co3 Wikipedia
    (Text) CC BY-SA