Dear all, > How can i generate a group e.g "Sp6(2)" as a matrix group by 7 * 7 matrices, > then generate "S8" as a matrix group inside Sp6(2) by 7 * 7 matrices using > GAP prog.
gap> G:=GeneralOrthogonalGroup(7,2); # use the fact that Sp(6,2)=O(7,2) GO(0,7,2) gap> o:=Orbits(G, GF(2)^7,OnLines);; gap> List(o,Size); # in this representation S8 fixes a hyperplane, not a vector [ 1, 63, 63, 1 ] gap> Gt:=Group(List(GeneratorsOfGroup(G),TransposedMat)); <matrix group with 2 generators> gap> ot:=Orbits(Gt, GF(2)^7,OnLines);; gap> List(ot,Length); # the stabiliser of a vector in 3rd orbit is S8 [ 1, 63, 36, 28 ] gap> s8:=Stabilizer(Gt,ot[3][1],OnLines); # here it is. <matrix group of size 40320 with 3 generators> Hope this helps Dima On Mon, Jun 29, 2020 at 07:32:51AM +0000, David Musyoka wrote: > Dear Forum, Dear Alexander Hulpke, > Given a group G, and and a vector space V of dimension n over GF(q), i am abe > to compute the orbit Lengths of V under action of G using the following GAP > commands: > V:=FullRowSpace(GF(q),n);Orb:=OrbitLengths(G,V) > My question is, how then do i compute the corresponding point stabilizers > (which are subgroups of G) for the orbits using GAP. > Thank you team in advance. > David. > Sent from Yahoo Mail on Android > > On Sun, 21 Jun 2020 at 22:47, David Musyoka<[email protected]> > wrote: Dear Alexander Hulpke,Dear Forum. > Thank You very much. > > Sent from Yahoo Mail on Android > > On Sun, 21 Jun 2020 at 19:48, > Hulpke,Alexander<[email protected]> wrote: Dear Forum, Dear > Marc David Musyoka, > > On Jun 20, 2020, at 00:22, David Musyoka <[email protected]> wrote: > Deat all, > Kind request to this team, i am new to GAP and i wish to be assisted in the > following, > How can i generate a group e.g "Sp6(2)" as a matrix group by 7 * 7 matrices, > then generate "S8" as a matrix group inside Sp6(2) by 7 * 7 matrices using > GAP prog. > > I wish that am assisted on how to execute the same step by step and the > matrix generators for the two groups be listed. > > > Yes, I also often wish that someone would assist me in every step and provide > me with the full result. > Anyhow, in this case (the algorithm is exponential time and attempts will > fail if groups are too large, or if the subgroup to be embedded needs many > generators) `IsomorphicSubgroups` seems to do the trick in a few minutes. > gap> G:=SP(6,2);Sp(6,2)gap> emb:=IsomorphicSubgroups(G,SymmetricGroup(8)); # > finds embeddings from S8 into G[ [ (1,8)(2,7)(3,5)(4,6), (1,8,4,7,3)(5,6) ] > -> [ <an immutable 6x6 matrix over GF2>, <an immutable 6x6 matrix over > GF2> ] ]gap> sub:=Image(emb[1]);<matrix group with 2 generators> > So there is one class of subgroups and `sub` is one (not necessarily the > nicest one) representative. > Of course this is computational overkill. The more sensible way would be to > produce the matrix representation (as reduced permutation representation), > find the form that it stabilizes, and then conjugate that form to the one > used for Sp. > Oh, here are the explicit matrix generators :-) > gap> GeneratorsOfGroup(sub);[ <an immutable 6x6 matrix over GF2>, <an > immutable 6x6 matrix over GF2> ] > (You can use `Print` or `Display` on each of them to see them with numbers.) > All the best, > Alexander Hulpke > -- Colorado State University, Department of Mathematics, > Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA > email: [email protected], > http://www.math.colostate.edu/~hulpke > > > > _______________________________________________ > Forum mailing list > [email protected] > https://mail.gap-system.org/mailman/listinfo/forum _______________________________________________ Forum mailing list [email protected] https://mail.gap-system.org/mailman/listinfo/forum
