Let $P$ be the set of all permutations in $S_{6}$ that can be expressed as a product of a multiple of $4$ transposition then $|P|$ is....$?$

Question

Let $P$ be the set of all permutations in $S_{6}$ that can be expressed as a product of a multiple of $4$ transposition then $|P|$ is...$?$

I counted all the permutations of $S_{6}$, that are product of $4$ disjoint cycles. These are $85$

But answer is given $360$. Maybe I'm not clearly getting, what the question is asking.

Where did I go wrong?