Conway group Co3

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For general background and history of the Conway sporadic groups, see Conway group.

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

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

History[edit]

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.

Representations[edit]

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.

Generalized Monstrous Moonshine[edit]

In analogy to monstrous moonshine for the monster M, for Co3, the relevant McKay-Thompson series is T_{4A}(\tau) where one can set the constant term a(0) = 24 (OEISA097340),

\begin{align}j_{4A}(\tau)
&=T_{4A}(\tau)+24\\
&=\Big(\tfrac{\eta^2(2\tau)}{\eta(\tau)\,\eta(4\tau)} \Big)^{24} \\
&=\Big(\big(\tfrac{\eta(\tau)}{\eta(4\tau)}\big)^{4}+4^2 \big(\tfrac{\eta(4\tau)}{\eta(\tau)}\big)^{4}\Big)^2\\
&=\frac{1}{q} + 24+ 276q + 2048q^2 +11202q^3+49152q^4+\dots
\end{align}

and η(τ) is the Dedekind eta function.

Maximal subgroups[edit]

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, 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, which moves 264 of the 276 type 2-2-3 triangles
  • [210.33]
  • S3 × PSL(2,8):3
  • A4 × S5

References[edit]

External links[edit]