# Conway group Co2

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vte :
In the area of modern algebra known as group theory, the Conway group Co$_2$ is a sporadic simple group of order
    42,305,421,312,000
    = 2$^{18}$ · 3$^6$ · 5$^3$ · 7 · 11 · 23
    ≈ 4×10$^{13}$.

## History and properties

Co$_2$ 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 Co$_0$. It is isomorphic to a subgroup of Co$_1$. The direct product 2×Co$_2$ is maximal in Co$_0$.
The Schur multiplier and the outer automorphism group are both trivial.

## Representations

Co$_2$ 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.
Co$_2$ 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 × Co$_2$ or Z/2Z × Co$_3$.
The Mathieu group M$_{23}$ is isomorphic to a maximal subgroup of Co$_2$ and one representation, in permutation matrices, fixes the type 2 vector u = (-3,1$^{23}$). 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 Co$_2$ has a convenient matrix representation inside the standard representation of Co$_0$. The trace of ζ is -8, while the involutions in M$_{23}$ have trace 8.
A 24-dimensional block sum of η and -η is in Co$_0$ if and only if the number of copies of η is odd.
Another representation fixes the vector v = (4,-4,0$^{22}$). A monomial and maximal subgroup includes a representation of M$_{22}$: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 Co$_2$ 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 Co$_2$.
There is an alternate way to construct the stabilizer of v. Now u and u+v = (1,-3,1$^{22}$) 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 M$_{22}$; these generate a group Fi$_{21}$ ≈ U$_6$(2). α (vide supra) extends this to Fi$_{21}$:2, which is maximal in Co$_2$. Lastly, Co$_0$ 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 Co$_2$ as follows:
Maximal subgroups of Co$_2$ :
| No. | Structure | Order | Index | Comments |
| 1 | Fi$_{21}$:2 ≈ U$_6$(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 Co$_2$ on 2300 such hexagons. Under this subgroup the hexagons are split into orbits of 1, 891, and 1408. Fi$_{21}$ fixes a 2-2-2 triangle defining the plane. |
| 2 | 2$^{10}$:M$_{22}$:2 | 908,328,960 = 2$^{18}$·3$^2$·5·7·11 | 46,575 = 3$^4$·5$^2$·23 | has monomial representation described above; 2$^{10}$:M$_{22}$ 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$^{1+8}$ +:Sp$_6$(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$^4$ × 2$^{1+6}$ +).A$_8$ | 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 | U$_4$(3):D$_8$ | 26,127,360 = 2$^{10}$·3$^6$·5·7 | 1,619,200 = 2$^8$·5$^2$·11·23 | |
| 8 | 2$^{4+10}$.(S$_5$ × S$_3$) | 11,796,480 = 2$^{18}$·3$^2$·5 | 3,586,275 = 3$^4$·5$^2$·7·11·23 | |
| 9 | M$_{23}$ | 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 | 3$^{1+4}$ +.2$^{1+4}$ $_–$.S$_5$ | 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$^{1+2}$ +:4S$_4$ | 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 Co$_2$ 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 Co$_2$ | 1 | 24 |
| 2A | 743,178,240 | 2$^{1+8}$:Sp$_6$(2) | 3$^2$·5$^2$·11·23 | -8 |
| 2B | 41,287,680 | 2$^{1+4}$:2$^4$.A$_8$ | 2·3$^4$·5$^2$11·23 | 8 |
| 2C | 1,474,560 | 2$^{10}$.A$_6$.2$^2$ | 2$^3$·3$^4$·5$^2$·7·11·23 | 0 |
| 3A | 466,560 | 3$^{1+4}$2$^{1+4}$A$_5$ | 2$^{11}$·5$^2$·7·11·23 | -3 |
| 3B | 155,520 | 3×U$_4$(2).2 | 2$^{11}$·3·5$^2$·7·11·23 | 6 |
| 4A | 3,096,576 | 4.2$^6$.U$_3$(3).2 | 2$^4$·3$^3$·5$^3$·11·23 | 8 |
| 4B | 122,880 | [2$^{10}$]S$_5$ | 2$^5$·3$^5$·5$^2$·7·11·23 | -4 |
| 4C | 73,728 | [2$^{13}$.3$^2$] | 2$^5$·3$^4$·5$^3$·7·11·23 | 4 |
| 4D | 49,152 | [2$^{14}$.3] | 2$^4$·3$^5$·5$^3$·7·11·23 | 0 |
| 4E | 6,144 | [2$^{11}$.3] | 2$^7$·3$^5$·5$^3$·7·11·23 | 4 |
| 4F | 6,144 | [2$^{11}$.3] | 2$^7$·3$^5$·5$^3$·7·11·23 | 0 |
| 4G | 1,280 | [2$^8$.5] | 2$^{10}$·3$^6$·5$^2$·7·11·23 | 0 |
| 5A | 3,000 | 5$^{1+2}$2A$_4$ | 2$^{15}$·3$^5$·7·11·23 | -1 |
| 5B | 600 | 5×S$_5$ | 2$^{15}$·3$^5$·5·7·11·23 | 4 |
| 6A | 5,760 | 3.2$^{1+4}$A5 | 2$^{11}$·3$^4$·5$^2$·7·11·23 | 5 |
| 6B | 5,184 | [2$^6$.3$^4$] | 2$^{12}$·3$^2$·5$^3$·7·11·23 | 1 |
| 6C | 4,320 | 6×S$_6$ | 2$^{13}$·3$^3$·5$^2$·7·11·23 | 4 |
| 6D | 3,456 | [2$^7$.3$^3$] | 2$^{11}$·3$^3$·5$^3$·7·11·23 | -2 |
| 6E | 576 | [2$^6$.3$^2$] | 2$^{12}$·3$^4$·5$^3$·7·11·23 | 2 |
| 6F | 288 | [2$^5$.3$^2$] | 2$^{13}$·3$^4$·5$^3$·7·11·23 | 0 |
| 7A | 56 | 7×D$_8$ | 2$^{15}$·3$^6$·5$^3$·11·233 | 3 |
| 8A | 768 | [2$^8$.3] | 2$^{10}$·3$^5$·5$^3$·7·11·23 | 0 |
| 8B | 768 | [2$^8$.3] | 2$^{10}$·3$^5$·5$^3$·7·11·23 | -2 |
| 8C | 512 | [2$^9$] | 2$^9$·3$^6$·5$^3$·7·11·23 | 4 |
| 8D | 512 | [2$^9$] | 2$^9$·3$^6$·5$^3$·7·11·23 | 0 |
| 8E | 256 | [2$^8$] | 2$^{10}$·3$^6$·5$^3$·7·11·23 | 2 |
| 8F | 64 | [2$^6$] | 2$^{12}$·3$^6$·5$^3$·7·11·23 | 2 |
| 9A | 54 | 9×S$_3$ | 2$^{17}$·3$^3$·5$^3$·7·11·23 | 3 |
| 10A | 120 | 5×2.A$_4$ | 2$^{15}$·3$^5$·5$^2$·7·11·23 | 3 |
| 10B | 60 | 10×S$_3$ | 2$^{16}$·3$^5$·5$^2$·7·11·23 | 2 |
| 10C | 40 | 5×D$_8$ | 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$^5$.3$^3$] | 2$^{13}$·3$^3$·5$^3$·7·11·23 | -1 |
| 12B | 288 | [2$^5$.3$^2$] | 2$^{13}$·3$^4$·5$^3$·7·11·23 | 1 |
| 12C | 288 | [2$^5$.3$^2$] | 2$^{13}$·3$^4$·5$^3$·7·11·23 | 2 |
| 12D | 288 | [2$^5$.3$^2$] | 2$^{13}$·3$^4$·5$^3$·7·11·23 | -2 |
| 12E | 96 | [2$^5$.3] | 2$^{13}$·3$^5$·5$^3$·7·11·23 | 3 |
| 12F | 96 | [2$^5$.3] | 2$^{13}$·3$^5$·5$^3$·7·11·23 | 2 |
| 12G | 48 | [2$^4$.3] | 2$^{14}$·3$^5$·5$^3$·7·11·23 | 1 |
| 12H | 48 | [2$^4$.3] | 2$^{14}$·3$^5$·5$^3$·7·11·23 | 0 |
| 14A | 56 | 5×D$_8$ | 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 | power equivalent |
| 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 | power equivalent |
| 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 | power equivalent |
| 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 |