The group determinants for ℤ_n × 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 $8$.