Injection of $D_6$ into $S_5.$

Question

Would anyone happen to recall how to put $D_6$ into $S_5?$ The cyclic generator has to go (up to conjugacy) to $(1 2 3) (4 5),$ but where does the involution go?

Answer 1

If you try $a\mapsto (123)(45)$, $b\mapsto (23)$, then the key relation $bab^{-1}=a^{-1}$ is satisfied.

Instead of $(23)$ you can use any other 2-cycle acting on $\{1,2,3\}$.

The key to success here is that $D_6\simeq D_3\times C_2$.