Conway group Co2 explained

In the area of modern algebra known as group theory, the Conway group Co₂ is a sporadic simple group of order

   42,305,421,312,000

= 2¹⁸3⁶5³71123

≈ 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 =

{ 1/2} \left ( \begin{matrix} 1 & -1 & -1 & -1 \\ -1 & 1 & -1 & -1 \\ -1 & -1 & 1 & -1 \\ -1 & -1 & -1 & 1 \end{matrix} \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:

Maximal subgroups of Co₂

No.	Structure	Order	Index	Comments
1	Fi21 2 ≈ U6(2):2	18,393,661,440 = 2^16·3^6·5·7·11	2,300 = 2^2·5^2·23	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.
2	2^10:M22:2	908,328,960 = 2^18·3^2·5·7·11	46,575 = 3^4·5^2·23	has monomial representation described above; 2^10:M22 fixes a 2-2-4 triangle.
3	McL	898,128,000 = 2^7·3^6·5^3·7·11	47,104 = 2^11·23	fixes a 2-2-3 triangle
4	2:Sp6(2)	743,178,240 = 2^18·3^4·5·7	56,925 = 3^2·5^2·11·23	centralizer of an involution of class 2A (trace -8)
5	HS 2	88,704,000 = 2^10·3^2·5^3·7·11	476,928 = 2^8·3^4·23	fixes a 2-3-3 triangle or exchanges its type 3 vertices with sign change
6	(2^4 ×).A8	41,287,680 = 2^17·3^2·5·7	1,024,650 = 2·3^4·5^2·11·23	centralizer of an involution of class 2B
7	U4(3):D8	26,127,360 = 2^10·3^6·5·7	1,619,200 = 2^8·5^2·11·23
8	2^4+10.(S5 × S3)	11,796,480 = 2^18·3^2·5	3,586,275 = 3^4·5^2·7·11·23
9	M23	10,200,960 = 2^7·3^2·5·7·11·23	4,147,200 = 2^11·3^4·5^2	fixes a 2-3-4 triangle
10	..S5	933,120 = 2^8·3^6·5	45,337,600 = 2^10·5^2·7·11·23	normalizer of a subgroup of order 3 (class 3A)
11	5:4S4	12,000 = 2^5·3·5^3	3,525,451,776 = 2^13·3^5·7·11·23	normalizer of a subgroup of order 5 (class 5A)

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. ^[1]

Centralizers of unknown structure are indicated with brackets.

Class	Order of centralizer	Centralizer	Size of class	Trace
1A		all Co2	1	24
2A	743,178,240	2^1+8:Sp6(2)	3^2·5^2·11·23	-8
2B	41,287,680	2^1+4:2^4.A8	2·3^4·5^211·23	8
2C	1,474,560	2^10.A6.2^2	2^3·3^4·5^2·7·11·23	0
3A	466,560	3^1+42^1+4A5	2^11·5^2·7·11·23	-3
3B	155,520	3×U4(2).2	2^11·3·5^2·7·11·23	6
4A	3,096,576	4.2^6.U3(3).2	2^4·3^3·5^3·11·23	8
4B	122,880	[2<sup>10</sup>]S5	2^5·3^5·5^2·7·11·23	-4
4C	73,728	[2<sup>13</sup>.3<sup>2</sup>]	2^5·3^4·5^3·7·11·23	4
4D	49,152	[2<sup>14</sup>.3]	2^4·3^5·5^3·7·11·23	0
4E	6,144	[2<sup>11</sup>.3]	2^7·3^5·5^3·7·11·23	4
4F	6,144	[2<sup>11</sup>.3]	2^7·3^5·5^3·7·11·23	0
4G	1,280	[2<sup>8</sup>.5]	2^10·3^6·5^2·7·11·23	0
5A	3,000	5^1+22A4	2^15·3^5·7·11·23	-1
5B	600	5×S5	2^15·3^5·5·7·11·23	4
6A	5,760	3.2^1+4A5	2^11·3^4·5^2·7·11·23	5
6B	5,184	[2<sup>6</sup>.3<sup>4</sup>]	2^12·3^2·5^3·7·11·23	1
6C	4,320	6×S6	2^13·3^3·5^2·7·11·23	4
6D	3,456	[2<sup>7</sup>.3<sup>3</sup>]	2^11·3^3·5^3·7·11·23	-2
6E	576	[2<sup>6</sup>.3<sup>2</sup>]	2^12·3^4·5^3·7·11·23	2
6F	288	[2<sup>5</sup>.3<sup>2</sup>]	2^13·3^4·5^3·7·11·23	0
7A	56	7×D8	2^15·3^6·5^3·11·233	3
8A	768	[2<sup>8</sup>.3]		2^10·3^5·5^3·7·11·23	0
8B	768	[2<sup>8</sup>.3]		2^10·3^5·5^3·7·11·23	-2
8C	512	[2<sup>9</sup>]		2^9·3^6·5^3·7·11·23	4
8D	512	[2<sup>9</sup>]	2^9·3^6·5^3·7·11·23	0
8E	256	[2<sup>8</sup>]	2^10·3^6·5^3·7·11·23	2
8F	64	[2<sup>6</sup>]	2^12·3^6·5^3·7·11·23	2
9A	54	9×S3	2^17·3^3·5^3·7·11·23	3
10A	120	5×2.A4	2^15·3^5·5^2·7·11·23	3
10B	60	10×S3	2^16·3^5·5^2·7·11·23	2
10C	40	5×D8	2^15·3^6·5^2·7·11·23	0
11A	11	11	2^18·3^6·5^3·7·23	2
12A	864	[2<sup>5</sup>.3<sup>3</sup>]	2^13·3^3·5^3·7·11·23	-1
12B	288	[2<sup>5</sup>.3<sup>2</sup>]	2^13·3^4·5^3·7·11·23	1
12C	288	[2<sup>5</sup>.3<sup>2</sup>]	2^13·3^4·5^3·7·11·23	2
12D	288	[2<sup>5</sup>.3<sup>2</sup>]	2^13·3^4·5^3·7·11·23	-2
12E	96	[2<sup>5</sup>.3]	2^13·3^5·5^3·7·11·23	3
12F	96	[2<sup>5</sup>.3]	2^13·3^5·5^3·7·11·23	2
12G	48	[2<sup>4</sup>.3]	2^14·3^5·5^3·7·11·23	1
12H	48	[2<sup>4</sup>.3]	2^14·3^5·5^3·7·11·23	0
14A	56	5×D8	2^15·3^6·5^3·11·23	-1
14B	28	14×2	2^16·3^6·5^3·11·23	1	power equivalent
14C	28	14×2	2^16·3^6·5^3·11·23	1
15A	30	30	2^17·3^5·5^2·7·11·23	1
15B	30	30	2^17·3^5·5^2·7·11·23	2	power equivalent
15C	30	30	2^17·3^5·5^2·7·11·23	2
16A	32	16×2	2^13·3^6·5^3·7·11·23	2
16B	32	16×2	2^13·3^6·5^3·7·11·23	0
18A	18	18	2^17·3^4·5^3·7·11·23	1
20A	20	20	2^16·3^6·5^2·7·11·23	1
20B	20	20	2^16·3^6·5^2·7·11·23	0
23A	23	23	2^18·3^6·5^3·7·11	1	power equivalent
23B	23	23	2^18·3^6·5^3·7·11	1
24A	24	24	2^15·3^5·5^3·7·11·23	0
24B	24	24	2^15·3^5·5^3·7·11·23	1
28A	28	28	2^16·3^6·5^3·11·23	1
30A	30	30	2^17·3^5·5^2·7·11·23	-1
30B	30	30	2^17·3^5·5^2·7·11·23	0
30C	30	30	2^17·3^5·5^2·7·11·23	0

References

• • • Reprinted in

Specific

1. Web site: ATLAS: Conway group Co2.

External links

• MathWorld: Conway Groups
• Atlas of Finite Group Representations: Co₂ version 2
• Atlas of Finite Group Representations: Co₂ version 3