Conway Group Co2

In the area of modern algebra known as group theory, the Conway group ''Co₂'' is a sporadic simple group of order
:   2¹⁸3⁶5³71123
: = 42305421312000
: ≈ 4.

History and properties

''Co₂'' is one of the 26 sporadic groups and was discovered by 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.

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 =
:$\left ( \begin 1 & -1 & -1 & -1 \\ -1 & 1 & -1 & -1 \\ -1 & -1 & 1 & -1 \\ -1 & -1 & -1 & 1 \end \right )$
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.

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. The names of conjugacy classes are taken from the Atlas of Finite Group Representations.

Centralizers of unknown structure are indicated with brackets.

References

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*{{Citation , last1=Wilson , first1=Robert A. , title=The finite simple groups. , publisher=Springer-Verlag , location=Berlin, New York , series=Graduate Texts in Mathematics 251 , isbn=978-1-84800-987-5 , doi=10.1007/978-1-84800-988-2 , zbl=1203.20012 , year=2009, volume=251

;Specific

External links

MathWorld: Conway Groups

Atlas of Finite Group Representations: Co₂
version 2

Atlas of Finite Group Representations: Co₂
version 3

Sporadic groups
John Horton Conway