FactorizeInGenerators := function(G,gens,e)
local i,n,F,hom,fgens;
n := Length(gens);
F :=FreeGroup(n);
fgens := GeneratorsOfGroup(F);
hom := GroupHomomorphismByImages(F,G,fgens,gens);
return PreImagesRepresentative(hom,e);
end;


#Z12
G := CyclicGroup(12);
Elements(G);
List(Elements(G),Order);
List(NormalSubgroups(G),Order);

g := (1,2,3,4,5,6,7,8,9,10,11,12);
G := Group(g);
g^2;
g^3;
Elements(G);
List([1..12], k->Order(g^k));


h := (2,12)(3,11)(4,10)(5,9)(6,8);
DIHED:= Group(g,h);
h^2;
g^12;
g*h;
h*g^(-1);


FactorizeInGenerators(DIHED,[g,h],g^4*h);
FactorizeInGenerators(DIHED,[g,h],(1,2,3));
NS := NormalSubgroups(DIHED);
List(NS, Order);

Elements(NS[2]);
List(Elements(NS[2]), e->FactorizeInGenerators(DIHED,[g,h],e));
List(Elements(NS[4]), e->FactorizeInGenerators(DIHED,[g,h],e));

KVOT := DIHED/NS[4];
List(Elements(KVOT),Order);

QQ := List(NS, Q -> DIHED/Q);
List(QQ,Order);

QQ[5];


List(Elements(QQ[5]),Order);



# Ny permgrupp

moves := [(1,2,3),(7,8,9),(1,4,7),(3,6,9)];
NG := Group(moves);
Order(NG);
Factorial(8)/Order(NG);
FactorizeInGenerators(NG,moves, (1,2)(7,8));
moves[1]*moves[3]^-1*moves[1]*moves[3]*moves[1]^-1*moves[3]^-1*moves[2]*moves[3]*moves[2]^-1*moves[3]^-1*moves[2]^-1