Is Symmetric group on 5 symbols is the semi-direct product?

Question

Is the Symmetric group on 5 symbols is the semi-direct product of groups $A_5$ and $C_2$, i.e. $$S_5\cong A_5\rtimes C_2?$$ Here $A_5$ is considered as a normal subgroup. Please help.

Answer 1

Yes indeed, we have a short exact sequence $1\to A_5\to S_5\to C_2\to1$ that "splits". The maps here are the natural ones. You can check that $\pi\circ i=\rm {id}_{C_2} $, where $i:C_2\hookrightarrow S_5$ is any of the $10$ inclusions.