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So I've been trying to shift a permutation group upwards to be able to make a cartesian product of two arbitrary permutation groups into a permutation group. My idea was to simply shift one permutation group up enough so that it commutes with the other. However I am stuck trying to figure out how to shift it up, i.e. add 5 to each number in the permutation. Anyone have any ideas?

Thanks

So I'm trying to take a permutation from a subgroup and add 5 to every digit in the permutation. For example if G=SymmetricGroup[5], G[2]=(1,2,3,4,5) , then I want the symmetric group isomorphic to G but where the 2nd element is (6,7,8,9,10). I also want to do this for arbitrary permutation groups, not just the symmetric groups.