Monolithic implies linearly primitive

Statement
Suppose $$G$$ is a finite group. Then, if $$G$$ is a fact about::monolithic group (i.e., it has a unique minimal normal subgroup) then $$G$$ is a fact about::linearly primitive group, i.e., it admits an irreducible linear representation over the complex numbers that is also a faithful linear representation, i.e., has trivial kernel).

Related facts

 * Linearly primitive implies cyclic-center