Cyclic permutation group example ($n>1$)

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I have googled around and haven't been able to find any examples of some $S_n$ with $n>1$ that is a cyclic group. This may mean it is a dumb question, any help is appreciated.

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A cyclic permutation group refers to a subgroup of $S_{n}$ that is cyclic. Examples here are easy. Just pick a single permutation and examine the group it generates.