Show $S_4$ can't be derived group 
Prove that there is no group $G$ such that $G' = S_4$.

We know that every group of order $24$ is not simple. and $S_4$ is an non-abelian group.
I tried to show that group of order $24$ can not be normal in any group (and because $G'$ is normal in $G, G'$ can not be $S_4$), but I can't prove that group of order $24$ can not be normal.

Write $C=C_G(G')$, then $C \cap G'=Z(S_4)=1$. By the N/C theorem $G/C$ injects homomorphically in ${\rm Aut}(S_4)=S_4$. Hence $(G/C)' =G'C/C \cong G'/(G' \cap C) \cong G'$ injects homomorphically in $(S_4)'=A_4$, which is impossible.
