Group distance magic graphs $G\times C_n$

A $\mathrm{\Gamma }$-distance magic labeling of a graph $G=(V,E)$ with $\mathrm{\mid }V\mathrm{\mid }=n$ is a bijection $f$ from $V$ to an Abelian group $\mathrm{\Gamma }$ of order $n$ such that the weight $w(x)={\sum }_{y\in N_G(x)}f(y)$ of every vertex $x\in V$ is equal to the same element $\mu \in \mathrm{\Gamma }$, called the \emph{magic constant}. In this paper we will show that if $G$ is a graph of order $n=2^p(2k+1)$ for some natural numbers $p$, $k$ such that $\mathrm{deg}(v)\equiv c\text{imod}{2}^{p+2}$ for some constant $c$ for any $v\in V(G)$, then there exists a $\mathrm{\Gamma }$-distance magic labeling for any Abelian group $\mathrm{\Gamma }$ of order $4n$ for the direct product $G\times C_4$. Moreover if $c$ is even then there exists a $\mathrm{\Gamma }$-distance magic labeling for any Abelian group $\mathrm{\Gamma }$ of order $8n$ for the direct product $G\times C_8$.