# What are the permutations in S7 that commute with (12)(345)?

I know that I have $(\sigma(1)\sigma(2))(\sigma(3)\sigma(4)\sigma(5))=(12)(345)$. However, I've only found eight that commute with this permutation, how do I know if I've found them all?

• Hint: the number elements is equal 7! / (the number of elements with the same cycle structure). – Steve D Nov 6 '17 at 3:59
• You haven't${}$. – Angina Seng Nov 6 '17 at 4:12
• Not sure what your notation $(\sigma(1)\sigma(2))(\sigma(3)\sigma(4)\sigma(5))=(12)(345)$ means unless you mean that $\sigma \in S_7$. Note that $H$, the subgroup generated by $(12)(345)$ is a cyclic group of order $6$. And the set of all permutations in $S_7$ that commute with $(12)(345)$ form another subgroup $K$ with $H \le K \le S_7$. This should give you an idea of what $|K|$ might be. – Stephen Meskin Nov 6 '17 at 5:14
• See this question, and several more of this type. – Dietrich Burde Nov 6 '17 at 20:22
• @SteveD So would it be 24 elements? – NoMayoPlz Nov 6 '17 at 22:59