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 =

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

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

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

References

• Conway, John Horton (1968), "A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups", Proceedings of the National Academy of Sciences of the United States of America, 61 (2): 398–400, Bibcode:1968PNAS...61..398C, doi:10.1073/pnas.61.2.398, MR 0237634, PMC 225171, PMID 16591697
• Conway, John Horton (1969), "A group of order 8,315,553,613,086,720,000", The Bulletin of the London Mathematical Society, 1: 79–88, doi:10.1112/blms/1.1.79, ISSN 0024-6093, MR 0248216
• Conway, John Horton (1971), "Three lectures on exceptional groups", in Powell, M. B.; Higman, Graham (eds.), Finite simple groups, Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969., Boston, MA: Academic Press, pp. 215–247, ISBN 978-0-12-563850-0, MR 0338152 Reprinted in Conway & Sloane (1999, 267–298)
• Conway, John Horton; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, vol. 290 (3rd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-2016-7, ISBN 978-0-387-98585-5, MR 0920369
• Feit, Walter (1974), "On integral representations of finite groups", Proceedings of the London Mathematical Society, Third Series, 29 (4): 633–683, doi:10.1112/plms/s3-29.4.633, ISSN 0024-6115, MR 0374248
• Thompson, Thomas M. (1983), From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs, vol. 21, Mathematical Association of America, ISBN 978-0-88385-023-7, MR 0749038
• Conway, John Horton; Parker, Richard A.; Norton, Simon P.; Curtis, R. T.; Wilson, Robert A. (1985), Atlas of finite groups, Oxford University Press, ISBN 978-0-19-853199-9, MR 0827219
• Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-03516-0, ISBN 978-3-540-62778-4, MR 1707296
• Wilson, Robert A. (1983), "The maximal subgroups of Conway's group ·2", Journal of Algebra, 84 (1): 107–114, doi:10.1016/0021-8693(83)90069-8, ISSN 0021-8693, MR 0716772
• Wilson, Robert A. (2009), The finite simple groups., Graduate Texts in Mathematics 251, vol. 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 1203.20012

Specific

External links

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