I got stuck with two questions dealing with characteristic groups

the first - G is a finite group characteristically simple and N is a minimal normal sub group in G (contains no proper normal subgroups of G). if $\displaystyle H=N_1 \times N_2 \times \cdots \times N_k$ is a direct sum of isomorphic copies of $\displaystyle N=N_1$ where k is maximal, then it is normal in G.

I can show that N is characteristically simple, and also all its isomorphic copies. I tried to show that if $\displaystyle gN_i g^{-1} \cap N_j \neq \{ e \}$ then $\displaystyle gN_i g^-1 \cap N_j = N_j$, and show that either there is another copy of N that I can join to the sum, or H is normal.

The problem is I can't see how I can use the fact that G is characteristically simple. I know that Z(G) and the commutator subgroups are characteristic in G but couldn't find what to do with it.

The second question is to show that if N is normal in G then $\displaystyle \Phi(N)\subseteq\Phi(G)$ where $\displaystyle \Phi(A)$ is the intersection of all maximal groups in A. I thougt to use $\displaystyle \Phi(G)\cap N$ and show that there is an intersection of some maximal groups in N that gives it. another thing is if H is maximal in G, and $\displaystyle \Phi(N)$ is not contained in H then maybe $\displaystyle H\Phi(G) $ will give a contradictory to the maximality of H (which is well defined bacuase $\displaystyle \Phi(G)$ char N normal in G).

any insights?