Some zero-sum problems over $⟨x,y\mid x^2=y^{n\mathrm{/}2},y^n=1,yx=x{y}^s⟩$

Let $n\ge 8$ be even, and let $G=⟨x,y\mid x^2=y^{n\mathrm{/}2},y^n=1,yx=x{y}^s⟩$, where $s^2\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}n)$ and $s\equiv ̸\pm 1(\mathrm{m}\mathrm{o}\mathrm{d}n)$. In this paper, we provide the precise values of some zero-sum constants over $G$, namely the small Davenport constant, $\eta$-constant, Gao constant, and Erd\H os-Ginzburg-Ziv constant. In particular, the Gao’s and Zhuang-Gao’s Conjectures hold for $G$. We also solve the associated inverse problems when $n\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$.