Matrix units in the simple components of rational group algebras

For the rational group algebra $\mathbb{Q}G$ of a finite group $G$, we provide an effective method to compute a complete set of matrix units and, in particular, primitive orthogonal idempotents in a simple component of $\mathbb{Q}G$, which is realized by a generalized strongly monomial character and has a prime Schur index. We also provide some classes of groups $G$ where this method can be successfully applied. The application of the method developed is also illustrated with detailed computations.