Prove that any finite group with exactly two minimal normal subgroups has a faithful irreducible $\mathbb{C}$-character.
What I have tried:
Let $N_1$ and $N_2$ be two minimal normal subgroups of $G$. If $G$ is abelian the result is straightforward. So let's suppose that $G'$ is nontrivial and then, one of the minimal normal subgroups, say $N_1$, is contained in $G'$. Now if $N_2 \cap G'=1$, there exist a $\lambda \in Lin(G)$ with $N_2 \cap Ker(\lambda)=1$ and also a nonlinear irreducible character $\phi$ such that $N_2 \subset Ker(\phi)$ and $N_1 \cap Ker(\phi)=1$. In this situation $\lambda \phi$ will be a faithful irreducible character of $G$. But what if both $N_1$ and $N_2$ are contained in $G'$? How should I deal with this situation?