The group determinants for ${\mathbb{Z}}_n\times H$

Let ${\mathbb{Z}}_n$ denote the cyclic group of order $n$. We show how the group determinant for $G={\mathbb{Z}}_n\times H$ can be simply written in terms of the group determinant for $H$. We use this to get a complete description of the integer group determinants for ${\mathbb{Z}}_2\times D_8$ where $D_8$ is the dihedral group of order 8, and ${\mathbb{Z}}_2\times Q_8$ where $Q_8$ is the quaternion group of order 8.