Conway group Co2

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 Co₂ is a sporadic simple group of order

   42,305,421,312,000

= 2¹⁸ · 3⁶ · 5³ · 7 · 11 · 23

≈ 4×10¹³.

History and properties

Co₂ 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 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.

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.

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 =

${\displaystyle {\mathbf {1} /2}\left({\begin{matrix}1&-1&-1&-1\\-1&1&-1&-1\\-1&-1&1&-1\\-1&-1&-1&1\end{matrix}}\right)}$

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.

Wilson (2009) 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 = 216·36·5·7·11	2,300 = 22·52·23	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	210:M22:2	908,328,960 = 218·32·5·7·11	46,575 = 34·52·23	has monomial representation described above; 210:M22 fixes a 2-2-4 triangle.
3	McL	898,128,000 = 27·36·53·7·11	47,104 = 211·23	fixes a 2-2-3 triangle
4	21+8 +:Sp6(2)	743,178,240 = 218·34·5·7	56,925 = 32·52·11·23	centralizer of an involution of class 2A (trace -8)
5	HS:2	88,704,000 = 210·32·53·7·11	476,928 = 28·34·23	fixes a 2-3-3 triangle or exchanges its type 3 vertices with sign change
6	(24 × 21+6 +).A8	41,287,680 = 217·32·5·7	1,024,650 = 2·34·52·11·23	centralizer of an involution of class 2B
7	U4(3):D8	26,127,360 = 210·36·5·7	1,619,200 = 28·52·11·23
8	24+10.(S5 × S3)	11,796,480 = 218·32·5	3,586,275 = 34·52·7·11·23
9	M23	10,200,960 = 27·32·5·7·11·23	4,147,200 = 211·34·52	fixes a 2-3-4 triangle
10	31+4 +.21+4  –.S5	933,120 = 28·36·5	45,337,600 = 210·52·7·11·23	normalizer of a subgroup of order 3 (class 3A)
11	51+2 +:4S4	12,000 = 25·3·53	3,525,451,776 = 213·35·7·11·23	normalizer of a subgroup of order 5 (class 5A)

Conjugacy classes

Traces of matrices in a standard 24-dimensional representation of Co₂ are shown.^[1] The names of conjugacy classes are taken from the Atlas of Finite Group Representations. ^[2]

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	21+8:Sp6(2)	32·52·11·23	-8
2B	41,287,680	21+4:24.A8	2·34·5211·23	8
2C	1,474,560	210.A6.22	23·34·52·7·11·23	0
3A	466,560	31+421+4A5	211·52·7·11·23	-3
3B	155,520	3×U4(2).2	211·3·52·7·11·23	6
4A	3,096,576	4.26.U3(3).2	24·33·53·11·23	8
4B	122,880	[210]S5	25·35·52·7·11·23	-4
4C	73,728	[213.32]	25·34·53·7·11·23	4
4D	49,152	[214.3]	24·35·53·7·11·23	0
4E	6,144	[211.3]	27·35·53·7·11·23	4
4F	6,144	[211.3]	27·35·53·7·11·23	0
4G	1,280	[28.5]	210·36·52·7·11·23	0
5A	3,000	51+22A4	215·35·7·11·23	-1
5B	600	5×S5	215·35·5·7·11·23	4
6A	5,760	3.21+4A5	211·34·52·7·11·23	5
6B	5,184	[26.34]	212·32·53·7·11·23	1
6C	4,320	6×S6	213·33·52·7·11·23	4
6D	3,456	[27.33]	211·33·53·7·11·23	-2
6E	576	[26.32]	212·34·53·7·11·23	2
6F	288	[25.32]	213·34·53·7·11·23	0
7A	56	7×D8	215·36·53·11·233	3
8A	768	[28.3]	210·35·53·7·11·23	0
8B	768	[28.3]	210·35·53·7·11·23	-2
8C	512	[29]	29·36·53·7·11·23	4
8D	512	[29]	29·36·53·7·11·23	0
8E	256	[28]	210·36·53·7·11·23	2
8F	64	[26]	212·36·53·7·11·23	2
9A	54	9×S3	217·33·53·7·11·23	3
10A	120	5×2.A4	215·35·52·7·11·23	3
10B	60	10×S3	216·35·52·7·11·23	2
10C	40	5×D8	215·36·52·7·11·23	0
11A	11	11	218·36·53·7·23	2
12A	864	[25.33]	213·33·53·7·11·23	-1
12B	288	[25.32]	213·34·53·7·11·23	1
12C	288	[25.32]	213·34·53·7·11·23	2
12D	288	[25.32]	213·34·53·7·11·23	-2
12E	96	[25.3]	213·35·53·7·11·23	3
12F	96	[25.3]	213·35·53·7·11·23	2
12G	48	[24.3]	214·35·53·7·11·23	1
12H	48	[24.3]	214·35·53·7·11·23	0
14A	56	5×D8	215·36·53·11·23	-1
14B	28	14×2	216·36·53·11·23	1	power equivalent
14C	28	14×2	216·36·53·11·23	1
15A	30	30	217·35·52·7·11·23	1
15B	30	30	217·35·52·7·11·23	2	power equivalent
15C	30	30	217·35·52·7·11·23	2
16A	32	16×2	213·36·53·7·11·23	2
16B	32	16×2	213·36·53·7·11·23	0
18A	18	18	217·34·53·7·11·23	1
20A	20	20	216·36·52·7·11·23	1
20B	20	20	216·36·52·7·11·23	0
23A	23	23	218·36·53·7·11	1	power equivalent
23B	23	23	218·36·53·7·11	1
24A	24	24	215·35·53·7·11·23	0
24B	24	24	215·35·53·7·11·23	1
28A	28	28	216·36·53·11·23	1
30A	30	30	217·35·52·7·11·23	-1
30B	30	30	217·35·52·7·11·23	0
30C	30	30	217·35·52·7·11·23	0