Permutations in $S_5$ commuting with (123)

Question

I know that $\tau \circ \sigma \circ \tau^{-1} = (\tau(1) \tau(2) \tau(3))$ for $\sigma = (123)$ but I don't know how to deduce everything that commutes with $(123)$ in $S_5$. Any hints/tips/solutions would be appreciated.

Thanks

Answer 1

Hint: $$\tau\circ\sigma=(\tau(1)\tau(2)\tau(3))\circ\tau\tag{1}$$ so it's enough to find all $\tau$ such that $(\tau(1)\tau(2)\tau(3))=(123)$. This gives the values of $\tau$ on the domain $\{1,2,3\}$, leaving $4,5$ free.