Let G be a finite group. An irreducible character χ is called monolithic when the factor group G/ker⁡(χ) has unique minimal normal subgroup. In this paper, we prove that for the smallest prime q dividing the order of G if G has a faithful imprimitive monolithic character of degree q, then G becomes a nonabelian q-group or a Frobenius group with cyclic Frobenius complement whose order is q. Under certain conditions, we also classify finite groups in which their nonlinear irreducible characters are monolithic.