# Conway group Co3

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In the area of modern algebra known as group theory, the Conway group $\mathrm {Co} _{3}$ is a sporadic simple group of order
    495,766,656,000
    = 2$^{10}$ · 3$^7$ · 5$^3$ · 7 · 11 · 23
    ≈ 5×10$^{11}$.

## History and properties

$\mathrm {Co} _{3}$ is one of the 26 sporadic groups and was discovered by John Horton Conway (1968, 1969) as the group of automorphisms of the Leech lattice $\Lambda$ fixing a lattice vector of type 3, thus length √6. It is thus a subgroup of $\mathrm {Co} _{0}$. It is isomorphic to a subgroup of $\mathrm {Co} _{1}$. The direct product $2\times \mathrm {Co} _{3}$ is maximal in $\mathrm {Co} _{0}$.
The Schur multiplier and the outer automorphism group are both trivial.

## Representations

Co$_3$ acts on the unique 23-dimensional even lattice of determinant 4 with no roots, given by the orthogonal complement of a norm 4 vector of the Leech lattice. This gives 23-dimensional representations over any field; over fields of characteristic 2 or 3 this can be reduced to a 22-dimensional faithful representation.
Co$_3$ has a doubly transitive permutation representation on 276 points.
Walter 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 $\mathbb {Z} /2\mathbb {Z} \times \mathrm {Co} _{2}$ or $\mathbb {Z} /2\mathbb {Z} \times \mathrm {Co} _{3}$.

## 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.
Larry Finkelstein (1973) found the 14 conjugacy classes of maximal subgroups of $\mathrm {Co} _{3}$ as follows:
Maximal subgroups of Co$_3$ :
| No. | Structure | Order | Index | Comments |
| 1 | McL:2 | 1,796,256,000 = 2$^8$·3$^6$·5$^3$·7·11 | 276 = 2$^2$·3·23 | McL fixes a 2-2-3 triangle. The maximal subgroup also includes reflections of the triangle. $\mathrm {Co} _{3}$ has a doubly transitive permutation representation on 276 type 2-2-3 triangles having as an edge a type 3 vector fixed by $\mathrm {Co} _{3}$. |
| 2 | HS | 44,352,000 = 2$^9$·3$^2$·5$^3$·7·11 | 11,178 = 2·3$^5$·23 | fixes a 2-3-3 triangle |
| 3 | U$_4$(3).2$^2$ | 13,063,680 = 2$^9$·3$^6$·5·7 | 37,950 = 2·3·5$^2$·11·23 | |
| 4 | M$_{23}$ | 10,200,960 = 2$^7$·3$^2$·5·7·11·23 | 48,600 = 2$^3$·3$^5$·5$^2$ | fixes a 2-3-4 triangle |
| 5 | 3$^5$:(2 × M$_{11}$) | 3,849,120 = 2$^5$·3$^7$·5·11 | 128,800 = 2$^5$·5$^2$·7·23 | fixes or reflects a 3-3-3 triangle |
| 6 | 2$^·$Sp$_6$(2) | 2,903,040 = 2$^{10}$·3$^4$·5·7 | 170,775 = 3$^3$·5$^2$·11·23 | centralizer of an involution of class 2A (trace 8), which moves 240 of the 276 type 2-2-3 triangles |
| 7 | U$_3$(5):S$_3$ | 756,000 = 2$^5$·3$^3$·5$^3$·7 | 655,776 = 2$^5$·3$^4$·11·23 | |
| 8 | 3$^{1+4}$ +:4S$_6$ | 699,840 = 2$^6$·3$^7$·5 | 708,400 = 2$^4$·5$^2$·7·11·23 | normalizer of a subgroup of order 3 (class 3A) |
| 9 | 2$^{4·}$A$_8$ | 322,560 = 2$^{10}$·3$^2$·5·7 | 1,536,975 = 3$^5$·5$^2$·11·23 | |
| 10 | PSL(3,4):(2 × S$_3$) | 241,920 = 2$^8$·3$^3$·5·7 | 2,049,300 = 2$^2$·3$^4$·5$^2$·11·23 | |
| 11 | 2 × M$_{12}$ | 190,080 = 2$^7$·3$^3$·5·11 | 2,608,200 = 2$^3$·3$^4$·5$^2$·7·23 | centralizer of an involution of class 2B (trace 0), which moves 264 of the 276 type 2-2-3 triangles |
| 12 | [2$^{10}$.3$^3$] | 27,648 = 2$^{10}$·3$^3$ | 17,931,375 = 3$^4$·5$^3$·7·11·23 | |
| 13 | S$_3$ × PSL(2,8):3 | 9,072 = 2$^4$·3$^4$·7 | 54,648,000 = 2$^6$·3$^3$·5$^3$·11·23 | normalizer of a subgroup of order 3 (class 3C, trace 0) |
| 14 | A$_4$ × S$_5$ | 1,440 = 2$^5$·3$^2$·5 | 344,282,400 = 2$^5$·3$^5$·5$^2$·7·11·23 | |

## Conjugacy classes

Traces of matrices in a standard 24-dimensional representation of Co$_3$ are shown. The names of conjugacy classes are taken from the Atlas of Finite Group Representations.
The cycle structures listed act on the 276 2-2-3 triangles that share the fixed type 3 side.
| Class | Order of centralizer | Size of class | Trace | Cycle type | |
| 1A | all Co$_3$ | 1 | 24 | |
| 2A | 2,903,040 | 3$^3$·5$^2$·11·23 | 8 | 1$^{36}$,2$^{120}$ |
| 2B | 190,080 | 2$^3$·3$^4$·5$^2$·7·23 | 0 | 1$^{12}$,2$^{132}$ |
| 3A | 349,920 | 2$^5$·5$^2$·7·11·23 | -3 | 1$^6$,3$^{90}$ |
| 3B | 29,160 | 2$^7$·3·5$^2$·7·11·23 | 6 | 1$^{15}$,3$^{87}$ |
| 3C | 4,536 | 2$^7$·3$^3$·5$^3$·11·23 | 0 | 3$^{92}$ |
| 4A | 23,040 | 2·3$^5$·5$^2$·7·11·23 | -4 | 1$^{16}$,2$^{10}$,4$^{60}$ |
| 4B | 1,536 | 2·3$^6$·5$^3$·7·11·23 | 4 | 1$^8$,2$^{14}$,4$^{60}$ |
| 5A | 1500 | 2$^8$·3$^6$·7·11·23 | -1 | 1,5$^{55}$ |
| 5B | 300 | 2$^8$·3$^6$·5·7·11·23 | 4 | 1$^6$,5$^{54}$ |
| 6A | 4,320 | 2$^5$·3$^4$·5$^2$·7·11·23 | 5 | 1$^6$,3$^{10}$,6$^{40}$ |
| 6B | 1,296 | 2$^6$·3$^3$·5$^3$·7·11·23 | -1 | 2$^3$,3$^{12}$,6$^{39}$ |
| 6C | 216 | 2$^7$·3$^4$·5$^3$·7·11·23 | 2 | 1$^3$,2$^6$,3$^{11}$,6$^{38}$ |
| 6D | 108 | 2$^8$·3$^4$·5$^3$·7·11·23 | 0 | 1$^3$,2$^6$,3$^3$,6$^{42}$ |
| 6E | 72 | 2$^7$·3$^5$·5$^3$·7·11·23 | 0 | 3$^4$,6$^{44}$ |
| 7A | 42 | 2$^9$·3$^6$·5$^3$·11·23 | 3 | 1$^3$,7$^{39}$ |
| 8A | 192 | 2$^4$·3$^6$·5$^3$·7·11·23 | 2 | 1$^2$,2$^3$,4$^7$,8$^{30}$ |
| 8B | 192 | 2$^4$·3$^6$·5$^3$·7·11·23 | -2 | 1$^6$,2,4$^7$,8$^{30}$ |
| 8C | 32 | 2$^5$·3$^7$·5$^3$·7·11·23 | 2 | 1$^2$,2$^3$,4$^7$,8$^{30}$ |
| 9A | 162 | 2$^9$·3$^3$·5$^3$·7·11·23 | 0 | 3$^2$,9$^{30}$ |
| 9B | 81 | 2$^{10}$·3$^3$·5$^3$·7·11·23 | 3 | 1$^3$,3,9$^{30}$ |
| 10A | 60 | 2$^8$·3$^6$·5$^2$·7·11·23 | 3 | 1,5$^7$,10$^{24}$ |
| 10B | 20 | 2$^8$·3$^7$·5$^2$·7·11·23 | 0 | 1$^2$,2$^2$,5$^2$,10$^{26}$ |
| 11A | 22 | 2$^9$·3$^7$·5$^3$·7·23 | 2 | 1,11$^{25}$ | power equivalent |
| 11B | 22 | 2$^9$·3$^7$·5$^3$·7·23 | 2 | 1,11$^{25}$ | power equivalent |
| 12A | 144 | 2$^6$·3$^5$·5$^3$·7·11·23 | -1 | 1$^4$,2,3$^4$,6$^3$,12$^{20}$ |
| 12B | 48 | 2$^6$·3$^6$·5$^3$·7·11·23 | 1 | 1$^2$,2$^2$,3$^2$,6$^4$,12$^{20}$ |
| 12C | 36 | 2$^8$·3$^5$·5$^3$·7·11·23 | 2 | 1,2,3$^5$,4$^3$,6$^3$,12$^{19}$ |
| 14A | 14 | 2$^9$·3$^7$·5$^3$·11·23 | 1 | 1,2,7$^5$14$^{17}$ |
| 15A | 15 | 2$^{10}$·3$^6$·5$^2$·7·11·23 | 2 | 1,5,15$^{18}$ |
| 15B | 30 | 2$^9$·3$^6$·5$^2$·7·11·23 | 1 | 3$^2$,5$^3$,15$^{17}$ |
| 18A | 18 | 2$^9$·3$^5$·5$^3$·7·11·23 | 2 | 6,9$^4$,18$^{13}$ |
| 20A | 20 | 2$^8$·3$^7$·5$^2$·7·11·23 | 1 | 1,5$^3$,10$^2$,20$^{12}$ | power equivalent |
| 20B | 20 | 2$^8$·3$^7$·5$^2$·7·11·23 | 1 | 1,5$^3$,10$^2$,20$^{12}$ | power equivalent |
| 21A | 21 | 2$^{10}$·3$^6$·5$^3$·11·23 | 0 | 3,21$^{13}$ |
| 22A | 22 | 2$^9$·3$^7$·5$^3$·7·23 | 0 | 1,11,22$^{12}$ | power equivalent |
| 22B | 22 | 2$^9$·3$^7$·5$^3$·7·23 | 0 | 1,11,22$^{12}$ | power equivalent |
| 23A | 23 | 2$^{10}$·3$^7$·5$^3$·7·11 | 1 | 23$^{12}$ | power equivalent |
| 23B | 23 | 2$^{10}$·3$^7$·5$^3$·7·11 | 1 | 23$^{12}$ | power equivalent |
| 24A | 24 | 2$^7$·3$^6$·5$^3$·7·11·23 | -1 | 1$^2$4,6,12$^2$24$^{10}$ |
| 24B | 24 | 2$^7$·3$^6$·5$^3$·7·11·23 | 1 | 2,3$^2$,4,12$^2$,24$^{10}$ |
| 30A | 30 | 2$^9$·3$^6$·5$^2$·7·11·23 | 0 | 1,5,15$^2$,30$^8$ |

## Generalized Monstrous Moonshine

In analogy to monstrous moonshine for the monster M, for Co$_3$, the relevant McKay-Thompson series is $T_{4A}(\tau )$ where one can set the constant term a(0) = 24 (OEIS: A097340),
    ${\begin{aligned}j_{4A}(\tau )&=T_{4A}(\tau )+24\\&={\Big{\tfrac {\eta ^{2}(2\tau )}{\eta (\tau )\,\eta (4\tau )}}{\Big )}^{24}\\&={\Big{\big{\tfrac {\eta (\tau )}{\eta (4\tau )}}{\big )}^{4}+4^{2}{\big{\tfrac {\eta (4\tau )}{\eta (\tau )}}{\big )}^{4}{\Big )}^{2}\\&={\frac {1}{q}}+24+276q+2048q^{2}+11202q^{3}+49152q^{4}+\dots \end{aligned}}$
and η(τ) is the Dedekind eta function.