Conway group Co₃

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In the area of modern algebra known as group theory, the Conway group ${\displaystyle \mathrm {Co} _{3}}$ is a

${\displaystyle \mathrm {Co} _{3}}$ is one of the 26 sporadic groups and was discovered by John Horton Conway (1968, 1969) as the group of automorphisms of the Leech lattice ${\displaystyle \Lambda }$ fixing a lattice vector of type 3, thus length √6. It is thus a subgroup of ${\displaystyle \mathrm {Co} _{0}}$. It is isomorphic to a subgroup of ${\displaystyle \mathrm {Co} _{1}}$. The direct product ${\displaystyle 2\times \mathrm {Co} _{3}}$ is maximal in ${\displaystyle \mathrm {Co} _{0}}$.

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.

Walter 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 ${\displaystyle \mathbb {Z} /2\mathbb {Z} \times \mathrm {Co} _{2}}$ or ${\displaystyle \mathbb {Z} /2\mathbb {Z} \times \mathrm {Co} _{3}}$.

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.

Larry Finkelstein (1973) found the 14 conjugacy classes of maximal subgroups of ${\displaystyle \mathrm {Co} _{3}}$ as follows:

• McL:2 – McL fixes a 2-2-3 triangle. The maximal subgroup also includes reflections of the triangle. ${\displaystyle \mathrm {Co} _{3}}$ has a
  doubly transitive permutation representation
  on 276 type 2-2-3 triangles having as an edge a type 3 vector fixed by ${\displaystyle \mathrm {Co} _{3}}$.
• HS – fixes a 2-3-3 triangle.
• U₄(3).2²
• M₂₃ – fixes a 2-3-4 triangle.
• 3⁵:(2 × M₁₁) - fixes or reflects a 3-3-3 triangle.
• 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 - normalizer of 3-subgroup generated by class 3C (trace 0) element
• A₄ × S₅

Conjugacy classes

Traces of matrices in a standard 24-dimensional representation of Co₃ are shown.^[1] The names of conjugacy classes are taken from the Atlas of Finite Group Representations.^{[2] [3]} The cycle structures listed act on the 276 2-2-3 triangles that share the fixed type 3 side.^[4]

Generalized Monstrous Moonshine

In analogy to monstrous moonshine for the monster M, for Co₃, the relevant McKay-Thompson series is ${\displaystyle T_{4A}(\tau )}$ where one can set the constant term a(0) = 24 (OEIS: A097340),

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

and η(τ) is the Dedekind eta function.

References

• PMID 16591697
• MR 0248216
• MR 0338152 Reprinted in Conway & Sloane (1999
  , 267–298)
• MR 0920369
• MR 0374248
• Finkelstein, Larry (1973), "The maximal subgroups of Conway's group C₃ and McLaughlin's group",
  MR 0346046
• Thompson, Thomas M. (1983), From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs, vol. 21,
  MR 0749038
• MR 0827219
• MR 1707296
• Wilson, Robert A. (2009), The finite simple groups., Graduate Texts in Mathematics 251, vol. 251, Berlin, New York: Springer-Verlag,
  Zbl 1203.20012