morphism between permutation group and matrix group
I can't figure out how to create a morphism between a permutation group and a matrix group (I know this is possible because Sage uses Gap and Gap doesn't have any problem with this, I don't imagine Sage removed the Gap functionality).
sage: flip = PermutationGroupElement("(1,2)") sage: g = PermutationGroup([flip]) sage: flop = Matrix(GF(3), 1, 1, ) sage: k = MatrixGroup([flop]) sage: g.order() 2 sage: k.order() 2 sage: g.hom([flop]) <snip useless error message> sage: k.hom([flip]) <snip another useless error message>
How can I construct the homomorphisms from g to k and k to g sending flip to flop and vice versa? (Maybe the problem is that Sage thinks g and k are a "permutation group" and "matrix group" respectively: if so, how can I get the underlying groups?)