The group determinants for ℤ_n × H

Let $\mathbb Z_n$ denote the cyclic group of order $n$. We show how determinant for $G= \mathbb Z_n \times H$ can be simply written in terms $H$. use this to get a complete description integer determinants Z_2 D_8$ where $D_8$ is dihedral $8$, and Q_8$ $Q_8$ quaternion $8$.