Let the finite group \(G\) possess a normal cyclic subgroup \(K\) such that \(G/K\) is cyclic. In that case, the subgroup \(A=C_G(K)\) is Abelian and normal in \(G\). Then \(G\) has a faithful irreducible representation over a field \(\mathbb{F}\) if and only if it satisfies one of the following conditions: (a) The characteristic of \(\mathbb{F}\) does not divide \(|A|\) and there exists a subgroup \(L