Conway Group Co3

In the area of modern algebra known as group theory, the Conway group ''$\mathrm_3$'' is a sporadic simple group of order
:   2¹⁰3⁷5³71123
: = 495766656000
: ≈ 5.

History and properties

''$\mathrm_3$'' is one of the 26 sporadic groups and was discovered by as the group of automorphisms of the Leech lattice $\Lambda$ fixing a lattice vector of type 3, thus length . It is thus a subgroup of $\mathrm_0$. It is isomorphic to a subgroup of $\mathrm_1$. The direct product $2\times \mathrm_3$ is maximal in $\mathrm_0$.

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.

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 $\Z/2\Z \times \mathrm_2$ or $\Z/2\Z \times \mathrm_3$.

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''.

found the 14 conjugacy classes of maximal subgroups of $\mathrm_3$ as follows:

* McL:2 – McL fixes a 2-2-3 triangle. The maximal subgroup also includes reflections of the triangle. $\mathrm_3$ has a doubly transitive permutation representation on 276 type 2-2-3 triangles having as an edge a type 3 vector fixed by $\mathrm_3$.
* 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
* ¹⁰.3³* 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. The names of conjugacy classes are taken from the Atlas of Finite Group Representations.

The cycle structures listed act on the 276 2-2-3 triangles that share the fixed type 3 side.

Generalized Monstrous Moonshine

In analogy to monstrous moonshine for the monster ''M'', for ''Co''₃, the relevant McKay-Thompson series is $T_(\tau)$ where one can set the constant term a(0) = 24 (),

:$\beginj_(\tau) &=T_(\tau)+24\\ &=\Big(\tfrac \Big)^ \\ &=\Big(\big(\tfrac\big)^+4^2 \big(\tfrac\big)^\Big)^2\\ &=\frac + 24+ 276q + 2048q^2 +11202q^3+49152q^4+\dots \end$

and ''η''(''τ'') is the Dedekind eta function.

References

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External links

MathWorld: Conway Groups

Atlas of Finite Group Representations: Co₃
version 2

Atlas of Finite Group Representations: Co₃
version 3

{{DEFAULTSORT:Conway Group Co3
Sporadic groups
John Horton Conway