Write $\left ( 123 \right )^{-1}\left ( 23 \right )\left ( 123 \right )$ in disjoint cycles.
I get that $\left ( 123 \right )^{-1}$=$\left ( 132 \right )$
$\left ( 132 \right )\left ( 23 \right )\left ( 123 \right )=\left ( 13 \right )\left ( 12 \right )\left ( 23 \right )\left ( 12 \right )\left ( 13 \right )$
Discounting the repeated cycles, I conclude the cycle is just $\left ( 23 \right )$. But this is wrong.
Any help is appreciated.