Abstract. Let [Formula: see text] be the cyclic group of order [Formula: see text] and [Formula: see text] be the dihedral group of order [Formula: see text], where [Formula: see text] is even. In this paper, we provide the exact values of some zero-sum constants over [Formula: see text], namely the small Davenport constant, Gao constant, [Formula: see text]–constant and Erdős–Ginzburg–Ziv constant. As a consequence, we prove Gao’s conjecture and the Zhuang–Gao conjecture for this group. These are the first concrete results on zero-sum problems for a family of non-abelian groups of rank greater than two.