find which of the following groups is isomorphic to S_{3}Z_{2}.

a) Z_{12}b) A_{4}c) D_{6}d) Z_{6}Z_{2}

I eliminate option a because Z12 is cyclic whereas S_{3}Z_{2}is not because we know that the External direct product of G and H is cyclic if and only if the orders of G and H are relatively prime. Here it's not the case.

Here's my question. Can I eliminate option d using the following argument?

If S_{3}Z_{2 }isomorphic to Z_{6}Z_{2}then we have S_{3}isomorphic to Z_{6}, which is again a contradiction as Z_{6}is cyclic whereas S_{3}is not.

Is my argument right?

Also it would be great if I can get a head start with the other options too...

Thanks