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We present a method to explicitly compute a complete set of orthogonalprimitive idempotents in a simple component with Schur index 1 of a rationalgroup algebra $\mathbb{Q}G$ for $G$ a finite generalized strongly monomialgroup. For the same groups with no exceptional simple components in$\mathbb{Q}G$, we describe a subgroup of finite index in the group of units$\mathcal{U}(\mathbb{Z}G)$ of the integral group ring $\mathbb{Z}G$ that isgenerated by three nilpotent groups for which we give explicit description oftheir generators. We exemplify the theoretical constructions with a detailedconcrete example to illustrate the theory. We also show that the Frobeniusgroups of odd order with a cyclic complement is a class of generalized stronglymonomial groups where the theory developed in this paper is applicable.

Rational group algebras of generalized strongly monomial groups: primitive idempotents and units