Erd\H{o}s-Ginzburg-Ziv theorem and Noether number for $C_m{⋉}_{\phi }{C}_{mn}$

Let $G$ be a multiplicative finite group and $S=a_1\cdot \dots \cdot a_k$ a sequence over $G$. We call $S$ a product-one sequence if $1={\prod }_{i=1}^k{a}_{\tau (i)}$ holds for some permutation $\tau$ of $\{1,\dots ,k\}$. The small Davenport constant $\mathsf{d}(G)$ is the maximal length of a product-one free sequence over $G$. For a subset $L\subset \mathbb N$,let$\mathsf s_L(G)$denotethesmallest$l\in\mathbb N_0\cup{\infty}$suchthateverysequence$S$over$G$oflength$|S|\ge l$ has a product-one subsequence $T$ of length $\mathrm{\mid }T\mathrm{\mid }\in L$. Denote $\mathsf e(G)=\max{$\text{ord}$(g): g\in G}$\mathrm{.}Someclassicalproduct-one(zero-sum)invariantsincluding$\mathsf D(G):=\mathsf s_{\mathbb N}(G)$(when$G$isabelian),$\mathsf E(G):=\mathsf s_{{|G|}}(G)$,$\mathsf s(G):=\mathsf s_{{\mathsf e(G)}}(G)$,$\eta(G):=\mathsf s_{[1,\mathsf e(G)]}(G)$and$\mathsf s_{d\mathbb N}(G)$($d\in\mathbb N$)havereceivedalotofstudies\mathrm{.}TheNoethernumber$\beta(G)$ which is closely related to zero-sum theory is defined to be the maximal degree bound for the generators of the algebra of polynomial invariants. Let $G\cong C_m\ltimes_{\varphi} C_{mn}$,inthispaper,weprovethat$$\mathsf E(G)=\mathsf d(G)+|G|=m^2n+m+mn-2$$and$\beta(G)=\mathsf d(G)+1=m+mn-1$. We also prove that ${\mathsf{s}}_{mn\mathbb{N}}(G)=m+2mn-2$ and provide the upper bounds of $\eta (G)$, $\mathsf{s}(G)$. Moreover, if $G$ is a non-cyclic nilpotent group and $p$ is the smallest prime divisor of $\mathrm{\mid }G\mathrm{\mid }$, we prove that $\beta (G)\le \frac{\mathrm{\mid }G\mathrm{\mid }}{p}+p-1$ except if $p=2$ and $G$ is a dicyclic group, in which case $\beta (G)=\frac{1}{2}\mathrm{\mid }G\mathrm{\mid }+2$.