We define a finite, solvable matrix group and convert it to a (general) polycyclic group G.
> a := GL(2,3) ! [1,1,0,1];
> b := GL(2,3) ! [0,1,1,0];
> M := sub<Parent(a)|a,b>;
> IsSolvable(M);
true
> IsFinite(M);
true
> G, f := GPCGroup(M);
We now compute the direct product D of G and the infinite dihedral group H, define a subgroup S of D, its normal closure N and construct the quotient Q of D by N.
> H<u,v> := DihedralGroup(GrpGPC, 0);
> D, incl, proj := DirectProduct(G, H);
> S := sub<D | incl[1](f(a*b)), incl[2]((u,v)^2)>;
> N := ncl<D|S>;
> Q := D/N;
> Q;
GrpGPC : Q of order 2^3 on 2 PC-generators
PC-Relations:
Q.1^2 = Id(Q),
Q.2^4 = Id(Q),
Q.2^Q.1 = Q.2^3
Since Q is finite, it can be transformed into a group of type
GrpPC using the function
PCGroup. This should (in non-trivial examples) be done, if further computations with it are intended.
> Q_ := PCGroup(Q);
> Q_;
GrpPC : Q_ of order 8 = 2^3
PC-Relations:
Q_.2^2 = Q_.3,
Q_.2^Q_.1 = Q_.2 * Q_.3
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