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

We present a method to explicitly compute a complete set of orthogonal primitive idempotents in a simple component with Schur index 1 of a rational group algebra QG for G a finite generalized strongly monomial group. For the same groups with no exceptional simple components in QG, we describe a subgroup of finite index in the group of units U(ZG) of the integral group ring ZG that is generated by three nilpotent groups for which we give explicit description of their generators. We exemplify the theoretical constructions with a detailed concrete example to illustrate the theory. We also show that the Frobenius groups of odd order with a cyclic complement are a class of generalized strongly monomial groups where the theory developed in this paper is applicable.