Wedderburn Decomposition and Idempotents of some finite metacyclic group algebras

In this article, we show explicitly the Wedderburn decomposition of the metacyclic group algebra ${\mathbb{F}}_{q}G$, where $G$ has a cyclic subgroup of index 2 and $\gcd (\mathrm{\mid }G\mathrm{\mid },q)=1$. We also construct the complete set of central and left idempotents of these group algebras.