For two groups to be isomorphic, there has to be a one-to-one mapping. This is impossible for $ S_4 $ and $ D_{12} $ because $ D_{12} $ has an order of 12, and $ S_4 $ does not,$ S_4=(14)(13)(12) $, but I could be wrong. -Dan
Yeah that's pretty much what I said. I put that elements of $ S_4 $ can only have orders of 1,2,3, or 4 and since $ D_{12} $ has an element of order 12, they are not isomorphic. -John
