Monolithic implies linearly primitive

Statement

Suppose $G$ is a finite group. Then, if $G$ is a Monolithic group (?) (i.e., it has a unique minimal normal subgroup) then $G$ is a 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