  Let $C_2$ be the cyclic group of order $2$ and $D_{2n}$ be the dihedral group
of order $2n$, where $n$ is even. In this paper, we provide the exact values of
some zero-sum constants over $D_{2n} \times C_2$, namely small Davenport
constant, Gao constant, $\eta$-constant and Erd\H os-Ginzburg-Ziv constant. As
a consequence, we prove the Gao's and Zhuang-Gao's Conjectures for this group.
These are the first concrete results on zero-sum problems for a family of
non-abelian groups of rank greater than $2$.
