What's the smallest $n$ such that $D_{12}$ has an isomorphic copy in $S_n$? I can show that $S_7$ is the smallest candidate for the property given. And, with a little calculation, I think it works out - $(1234)(567)$ and $(14)(23)(57)$ seem to generate such a subgroup.  But I was wondering if there's a more clever way of approaching the problem.  If so, is there a general strategy that works for a given $D_m$?

Every dihedral group is generated by two elements of order 2.  And every group generated by two elements of order 2 is dihedral.  Furthermore the order of the generated dihedral group is the order of the product of the two elements.  So if x,y each have order 2, then  is dihedral with 2*n elements where n is the order of x*y.  This is why dihedral groups show up so often in finite group theory, anytime you have a group with a lot of elements of order 2, you have a lot of dihedral subgroups.
A permutation has order 2 only if it is a product of 2-cycles, so it looks like (ab)(cd)(ef)...  So your question is equivalent to "What is the smallest n such that there are two products of 2-cycle swaps on n elements, and the product of those two permutations has order 12?"  There's still a search involved, but its a smaller set of permutations to search.
