[GAP Forum] Co1 generators in SO24

[Mathieu Dutour]

There are a number of ways to do that.

If you are interested in the automorphism group of the Leech lattice, that
is the double cover of the group Co1, then you can use my package
"polyhedral" (from http://www.liga.ens.fr/~dutour/Polyhedral/index.html)
which is not official or the package "cryst" which is official.
Both rely on the use of autom by B. Souvignier and W. Plesken and Magma
rely as well on this program.

But I should point to you that what you are asking is Co1, i.e. the quotient
of the automorphism group of the Leech lattice by the antipodal involution.
The atlas of finite groups http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co1/
does not list obvious 24-dimensional rational representations of this group.

  Mathieu

>> Hello,
>> 
>> I have received following email from one matematician. I have asked him for the matrix
>> generators of Conway group Co1 in SO(24). Do you know how to obtain such generators in GAP ? 
>> 
>> <quote>
>> The following Magma code should work:
>> 
>> L := Lattice("Lambda",24);
>> G := AutomorphismGroup(L);
>> B := BasisMatrix(L);
>> S := ShortestVectors(L);
>> S := S cat [-S[i] : i in [1..#S]];
>> M := MatrixRing(Rationals(),24);
>> G := MatrixGroup<24, Rationals() | [B^(-1) * M!G.i * B : i in [1..Ngens(G)]]>;
>> 
>> Then S will be the list of minimal vectors and G will be the
>> automorphism group, as a subgroup of SO(24). The code for G is
>> a little ugly, because by default Magma will express it as a
>> subgroup of GL_24(Z) instead.
>> < end of quote>
>> 
>> Here is the base matrix of my leech lattice. The determinant is 8^12.
>> B:=[[4,-4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
>> [4,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
>> [4,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
>> [4,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
>> [4,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
>> [4,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
>> [4,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
>> [2,2,2,2,2,2,2,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
>> [4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
>> [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
>> [4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0],
>> [2,2,2,2,0,0,0,0,2,2,2,2,0,0,0,0,0,0,0,0,0,0,0,0],
>> [4,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0],
>> [2,2,0,0,2,2,0,0,2,2,0,0,2,2,0,0,0,0,0,0,0,0,0,0],
>> [2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,0,0,0,0,0,0,0,0],
>> [2,0,0,2,2,0,0,2,2,0,0,2,2,0,0,2,0,0,0,0,0,0,0,0],
>> [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0],
>> [2,0,2,0,2,0,0,2,2,2,0,0,0,0,0,0,0,0,0,0,0,0,2,2],
>> [2,0,0,2,2,2,0,0,2,0,2,0,0,0,0,0,0,0,0,0,0,2,0,2],
>> [2,2,0,0,2,0,2,0,2,0,0,2,0,0,0,0,0,0,0,0,2,0,0,2],
>> [0,2,2,2,2,0,0,0,2,0,0,0,2,0,0,0,0,0,0,2,0,0,0,2],
>> [0,0,0,0,0,0,0,0,2,2,0,0,2,2,0,0,2,2,0,0,2,2,0,0],
>> [0,0,0,0,0,0,0,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0],
>> [-3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]];
>> 
>> Regards,
>> Marek
y