# 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 $D_m$?

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In your notation, is $D_{12}$ the symmetry group of a hexagon or a dodecagon? – Henning Makholm Dec 13 '13 at 22:24
What you are looking for is called a "permutation representation" of the dihedral group. You might find something, searching for those phrases. – Gerry Myerson Dec 14 '13 at 3:57
Sorry, Henning. It's the dodecagon. Is $D_{24}$ the preferred notation? Also, thanks for the guidance, Gerry. I'll read up on it. – Josh Keneda Dec 14 '13 at 12:39
Finding the smallest degree faithful permutation representation of a given finite group is a difficult problem in general, bit I expect it is possible to do this for dihedral groups. – Derek Holt Dec 14 '13 at 13:33
I think it's $\mathcal{S}_5.$ – Ivan Dec 14 '13 at 13:43