Group distance magic graphs $G\times C_n$

A $\Gamma$-distance magic labeling of a graph $G=(V,E)$ with $|V | = n$ is a bijection $f$ from $V$ to an Abelian group $\Gamma$ 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 $\mu \in \Gamma$, 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 $\deg(v)\equiv c \imod {2^{p+2}}$ for some constant $c$ for any $v\in V(G)$, then there exists a $\Gamma$-distance magic labeling for any Abelian group $\Gamma$ of order $4n$ for the direct product $G\times C_4$. Moreover if $c$ is even then there exists a $\Gamma$-distance magic labeling for any Abelian group $\Gamma$ of order $8n$ for the direct product $G\times C_8$.