minimal normal subgroups; faithful characters; strong condition on normal subgroups; Frobenius groups
In this paper, we consider finite groups with precisely one nonlinear nonfaithful irreducible character. We show that the groups of order 16 with nilpotency class 3 are the only $p$-groups with this property. Moreover we completely characterize the nilpotent groups with this property. Also we show that if $G$ is a group with a nontrivial center which possesses precisely one nonlinear nonfaithful irreducible character then $G$ is solvable.
 Di Martino, L., Tamburini, M. C.: Some remarks on the degrees of faithful irreducible representation of a finite group
. Geom. Dedicata 41 (1992), 155–164. MR 1153979
 Fernández–Alcober, G. A., Moretó, A.: Groups with extreme character degrees and their normal subgroups
. Trans. Amer. Math. Soc. 353 (2001), 2271–2292. DOI 10.1090/S0002-9947-01-02685-X
GAP Groups, Algorithms, and Programming, Version 4.4.10, 2007.