Isn't the obvious place the book SPLAG by Conway & Sloane? Chapter 4? Rob WIlson at QMUL is master of the generators. He has a new book on the finite simples.
John McKay == On Fri, 9 Apr 2010, Mathieu Dutour wrote: > 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 > > _______________________________________________ > Forum mailing list > Forum@mail.gap-system.org > http://mail.gap-system.org/mailman/listinfo/forum > _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
