Computing the cycle decomposition of the composition of permutations. Let $\sigma,\tau\in S_5$ be given by
$$
\sigma = \begin{pmatrix}1&3&5\end{pmatrix}\begin{pmatrix}2&4\end{pmatrix},\quad \tau = \begin{pmatrix}1&5 \end{pmatrix}\begin{pmatrix}2&3\end{pmatrix}.
$$
I want to find the cycle decomposition of $\tau\sigma$. So I write
$$
\tau\sigma = \begin{pmatrix}1&5 \end{pmatrix}\begin{pmatrix}2&3\end{pmatrix}\begin{pmatrix}1&3&5\end{pmatrix}\begin{pmatrix}2&4\end{pmatrix}.
$$
Reading from right to left and starting with the element $1$, I see that $1$ is taken to $3$ and $3$ is taken to $2$. Now, $2$ is taken to $4$, and $4$ is fixed in the other cycles, so our first cycle is $\begin{pmatrix}1&2&4\end{pmatrix}$. Since $3$ is taken to $5$ and $5$ is taken to $1$, the next cycle would be $\begin{pmatrix}3&1\end{pmatrix}$. But these cycles aren't disjoint, so I must have done something wrong. Where is my error?

$4$ is taken to $2$ and $2$ is taken to $3$, so it's $(1243)$.
