Character degrees of nilpotent-by-metacyclic groups (Q1320217)

Let \(G\) be a finite solvable group and let \(\pi(n)\) be the set of prime divisors of an integer \(n\). Let \[ \rho(G) = \bigcup_{\chi \in \text{Irr}(G)} \pi(\chi(1)) \quad \text{and}\quad \sigma(G) = \max_{\chi\in \text{Irr}(G)} | \pi(\chi(1))|. \] A question of Huppert concerns the validity of the inequality \(| \rho(G)| \leq 2\sigma (G)\). Gluck and Manz have shown that this problem may be reduced to the nilpotent-by-metabelian case. The author proves that if \(G\) is nilpotent-by-metacyclic, then \(| \rho(G)| \leq 2 \sigma(G)\).