Monolithic finite group has faithful irreducible representation

Statement
Suppose $$G$$ is a finite group that is a monolithic group, i.e., it has a unique minimal normal subgroup that is contained in every nontrivial normal subgroup. Suppose $$K$$ is a splitting field for $$G$$. Then, $$G$$ has a faithful irreducible representation over $$K$$. In particular, $$G$$ is a fact about::linearly primitive group.