# Conway group Co1

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vte :
In the area of modern algebra known as group theory, the Conway group Co$_1$ is a sporadic simple group of order
    4,157,776,806,543,360,000
    = 2$^{21}$ · 3$^9$ · 5$^4$ · 7$^2$ · 11 · 13 · 23
    ≈ 4×10$^{18}$.

## History and properties

Co$_1$ is one of the 26 sporadic groups and was discovered by John Horton Conway in 1968. It is the largest of the three sporadic Conway groups and can be obtained as the quotient of Co$_0$ (group of automorphisms of the Leech lattice Λ that fix the origin) by its center, which consists of the scalar matrices ±1. It also appears at the top of the automorphism group of the even 26-dimensional unimodular lattice II$_{25,1}$. Some rather cryptic comments in Witt's collected works suggest that he found the Leech lattice and possibly the order of its automorphism group in unpublished work in 1940.
The outer automorphism group is trivial and the Schur multiplier has order 2.

## Involutions

Co$_0$ has 4 conjugacy classes of involutions; these collapse to 2 in Co$_1$, but there are 4-elements in Co$_0$ that correspond to a third class of involutions in Co$_1$.
An image of a dodecad has a centralizer of type 2$^{11}$:M$_{12}$:2, which is contained in a maximal subgroup of type 2$^{11}$:M$_{24}$.
An image of an octad or 16-set has a centralizer of the form 2$^{1+8}$.O$^+$ $_8$(2), a maximal subgroup.

## Representations

The smallest faithful permutation representation of Co$_1$ is on the 98280 pairs {v,–v} of norm 4 vectors.
There is a matrix representation of dimension 24 over the field $\mathbb {F} _{2}$.
The centralizer of an involution of type 2B in the monster group is of the form 2$^{1+24}$Co$_1$.
The Dynkin diagram of the even Lorentzian unimodular lattice II$_{1,25}$ is isometric to the (affine) Leech lattice Λ, so the group of diagram automorphisms is split extension Λ,Co$_0$ of affine isometries of the Leech lattice.

## Maximal subgroups

Wilson (1983) found the 22 conjugacy classes of maximal subgroups of Co$_1$, though there were some errors in this list, corrected by Wilson (1988).
Maximal subgroups of Co$_1$ :
| No. | Structure | Order | Index | Comments |
| 1 | Co$_2$ | 42,305,421,312,000 = 2$^{18}$·3$^6$·5$^3$·7·11·23 | 98,280 = 2$^3$·3$^3$·5·7·13 | |
| 2 | 3$^·$Suz:2 | 2,690,072,985,600 = 2$^{14}$·3$^8$·5$^2$·7·11·13 | 1,545,600 = 2$^7$·3·5$^2$·7·23 | the lift to Aut(Λ) = Co$_0$ fixes a complex structure or changes it to the complex conjugate structure; also, top of Suzuki chain |
| 3 | 2$^{11}$:M$_{24}$ | 501,397,585,920 = 2$^{21}$·3$^3$·5·7·11·23 | 8,292,375 = 3$^6$·5$^3$·7·13 | image of monomial subgroup from Aut(Λ), that subgroup stabilizing the standard frame of 48 vectors of form (±8,0$^{23}$) |
| 4 | Co$_3$ | 495,766,656,000 = 2$^{10}$·3$^7$·5$^3$·7·11·23 | 8,386,560 = 2$^{11}$·3$^2$·5·7·13 | |
| 5 | 2$^{1+8·}$O$^+$ $_8$(2) | 89,181,388,800 = 2$^{21}$·3$^5$·5$^2$·7 | 46,621,575 = 3$^4$·5$^2$·7·11·13·23 | centralizer of an involution of class 2A (image of octad from Aut(Λ)) |
| 6 | Fi$_{21}$:S$_3$ ≈ U$_6$(2):S$_3$ | 55,180,984,320 = 2$^{16}$·3$^7$·5·7·11 | 75,348,000 = 2$^5$·3$^2$·5$^3$·7·13·23 | the lift to Aut(Λ) is the symmetry group of a coplanar hexagon of 6 type 2 points |
| 7 | (A$_4$ × G$_2$(4)):2 | 6,038,323,200 = 2$^{15}$·3$^4$·5$^2$·7·13 | 688,564,800 = 2$^6$·3$^5$·5$^2$·7·11·23 | in Suzuki chain |
| 8 | 2$^{2+12}$:(A$_8$ × S$_3$) | 1,981,808,640 = 2$^{21}$·3$^3$·5·7 | 2,097,970,875 = 3$^6$·5$^3$·7·11·13·23 | |
| 9 | 2$^{4+12·}$(S$_3$ × 3.S$_6$) | 849,346,560 = 2$^{21}$·3$^4$·5 | 4,895,265,375 = 3$^5$·5$^3$·7$^2$·11·13·23 | |
| 10 | 3$^{2·}$U$_4$(3).D$_8$ | 235,146,240 = 2$^{10}$·3$^8$·5·7 | 17,681,664,000 = 2$^{11}$·3·5$^3$·7·11·13·23 | |
| 11 | 3$^6$:2.M$_{12}$ | 138,568,320 = 2$^7$·3$^9$·5·11 | 30,005,248,000 = 2$^{14}$·5$^3$·7$^2$·13·23 | holomorph of ternary Golay code |
| 12 | (A$_5$ × J$_2$):2 | 72,576,000 = 2$^{10}$·3$^4$·5$^3$·7 | 57,288,591,360 = 2$^{11}$·3$^5$·5·7·11·13·23 | in Suzuki chain |
| 13 | 3$^{1+4}$:2.S$_4$(3).2 | 25,194,240 = 2$^8$·3$^9$·5 | 165,028,864,000 = 2$^{13}$·5$^3$·7$^2$·11·13·23 | |
| 14 | (A$_6$ × U$_3$(3)).2 | 4,354,560 = 2$^9$·3$^5$·5·7 | 954,809,856,000 = 2$^{12}$·3$^4$·5$^3$·7·11·13·23 | in Suzuki chain |
| 15 | 3$^{3+4}$:2.(S$_4$ × S$_4$) | 2,519,424 = 2$^7$·3$^9$ | 1,650,288,640,000 = 2$^{14}$·5$^4$·7$^2$·11·13·23 | |
| 16 | A$_9$ × S$_3$ | 1,088,640 = 2$^7$·3$^5$·5·7 | 3,819,239,424,000 = 2$^{14}$·3$^4$·5$^3$·7·11·13·23 | in Suzuki chain |
| 17 | (A$_7$ × L$_2$(7)):2 | 846,720 = 2$^7$·3$^3$·5·7$^2$ | 4,910,450,688,000 = 2$^{14}$·3$^6$·5$^3$·11·13·23 | in Suzuki chain |
| 18 | (D$_{10}$ × (A$_5$ × A$_5$).2).2 | 144,000 = 2$^7$·3$^2$·5$^3$ | 28,873,450,045,440 = 2$^{14}$·3$^7$·5·7$^2$·11·13·23 | |
| 19 | 5$^{1+2}$:GL$_2$(5) | 60,000 = 2$^5$·3·5$^4$ | 69,296,280,109,056 = 2$^{16}$·3$^8$·7$^2$·11·13·23 | |
| 20 | 5$^3$:(4 × A$_5$).2 | 60,000 = 2$^5$·3·5$^4$ | 69,296,280,109,056 = 2$^{16}$·3$^8$·7$^2$·11·13·23 | |
| 21 | 7$^2$:(3 × 2.S$_4$) | 3,528 = 2$^3$·3$^2$·7$^2$ | 1,178,508,165,120,000 = 2$^{18}$·3$^7$·5$^4$·11·13·23 | |
| 22 | 5$^2$:2A$_5$ | 3,000 = 2$^3$·3·5$^3$ | 1,385,925,602,181,120 = 2$^{18}$·3$^8$·5·7$^2$·11·13·23 | |