Prove $$\Phi(G)=G'G^p,$$ where $\Phi(G)$ is the Frattini subgroup of $G$, the intersection of all maximum subgroups of $G$, $G'=[G:G]$ is the commutator subgroup of $G$, and $G^p$ is the group generated by all $p$th powers of $G$.
I don't understand why this is true.