Conway group Co2

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 ·· 11 · 23
≈ 4×1013.

History and properties

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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

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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 =

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

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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

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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

References

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Specific
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