Conway group Co2

Co₂ is one of the 26 sporadic groups and was discovered by {{harvs |authorlink=John Horton Conway |last=Conway |year1=1968 |year2=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.

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.

{{harvtxt|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.

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.

{{harvtxt|Wilson|2009}} found the 11 conjugacy classes of maximal subgroups of Co₂ as follows:

class="wikitable" |+ Maximal subgroups of Co2
No.	Structure	Order	Index	Comments
1	Fi21:2 ≈ U6(2):2	style="text-align:right;"|18,393,661,440 = 216·36·5·7·11	align=right| 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	style="text-align:right;"| 908,328,960 = 218·32·5·7·11	align=right| 46,575 = 34·52·23	has monomial representation described above; 210:M22 fixes a 2-2-4 triangle.
3	McL	style="text-align:right;"| 898,128,000 = 27·36·53·7·11	align=right| 47,104 = 211·23	fixes a 2-2-3 triangle
4	2{{su|a=l|b=+|p=1+8}}:Sp6(2)	style="text-align:right;"| 743,178,240 = 218·34·5·7	align=right| 56,925 = 32·52·11·23	centralizer of an involution of class 2A (trace -8)
5	HS:2	style="text-align:right;"| 88,704,000 = 210·32·53·7·11	align=right| 476,928 = 28·34·23	fixes a 2-3-3 triangle or exchanges its type 3 vertices with sign change
6	(24 × {{nowrap|2{{su|a=l|b=+|p=1+6}}}}).A8	style="text-align:right;"| 41,287,680 = 217·32·5·7	align=right| 1,024,650 = 2·34·52·11·23	centralizer of an involution of class 2B
7	U4(3):D8	style="text-align:right;"| 26,127,360 = 210·36·5·7	align=right| 1,619,200 = 28·52·11·23
8	24+10.(S5 × S3)	style="text-align:right;"| 11,796,480 = 218·32·5	align=right| 3,586,275 = 34·52·7·11·23
9	M23	style="text-align:right;"| 10,200,960 = 27·32·5·7·11·23	align=right| 4,147,200 = 211·34·52	fixes a 2-3-4 triangle
10	{{nowrap|3{{su|a=l|b=+|p=1+4}}}}.{{nowrap|2{{su|a=l|b= –|p=1+4}}}}.S5	style="text-align:right;"| 933,120 = 28·36·5	align=right| 45,337,600 = 210·52·7·11·23	normalizer of a subgroup of order 3 (class 3A)
11	5{{su|a=l|b=+|p=1+2}}:4S4	style="text-align:right;"| 12,000 = 25·3·53	align=right|3,525,451,776 = 213·35·7·11·23	normalizer of a subgroup of order 5 (class 5A)

Traces of matrices in a standard 24-dimensional representation of Co₂ are shown.{{harvtxt|Wilson|1983}} The names of conjugacy classes are taken from the Atlas of Finite Group Representations. {{Cite web|url=http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co2/#ccls|title=ATLAS: Conway group Co2}}

Centralizers of unknown structure are indicated with brackets.

class="wikitable" style="margin: 1em auto;"
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	rowspan = "2"| 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	rowspan = "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	rowspan = "2"| 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

• {{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | title=A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups | mr=0237634 | year=1968 | journal=Proceedings of the National Academy of Sciences of the United States of America | volume=61 | pages=398–400 | doi=10.1073/pnas.61.2.398 | issue=2| pmc=225171 | pmid=16591697| bibcode=1968PNAS...61..398C | doi-access=free }}
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• {{Citation | last1=Thompson | first1=Thomas M. | title=From error-correcting codes through sphere packings to simple groups | url=https://books.google.com/books?id=ggqxuG31B3cC | publisher=Mathematical Association of America | series=Carus Mathematical Monographs | isbn=978-0-88385-023-7 | mr=749038 | year=1983 | volume=21}}
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• {{Citation | last1=Griess | first1=Robert L. Jr. | author1-link=R. L. Griess | title=Twelve sporadic groups | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-62778-4 | mr=1707296 | year=1998 | doi=10.1007/978-3-662-03516-0}}
• {{Citation | last1=Wilson | first1=Robert A. | title=The maximal subgroups of Conway's group ·2 | doi=10.1016/0021-8693(83)90069-8 | mr=716772 | year=1983 | journal=Journal of Algebra | issn=0021-8693 | volume=84 | issue=1 | pages=107–114| doi-access= }}
• {{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

• [http://mathworld.wolfram.com/ConwayGroups.html MathWorld: Conway Groups]
• [http://web.mat.bham.ac.uk/atlas/v2.0/spor/Co2/ Atlas of Finite Group Representations: Co₂] version 2
• [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co2/ Atlas of Finite Group Representations: Co₂] version 3

Category:Sporadic groups

Category:John Horton Conway