A group distance magic labeling or a \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{G}}$$\end{document} -distance magic labeling of a graph G = (V, E) with \documentclass[12pt]{minimal}
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\begin{document}$${|V | = n}$$\end{document} is a bijection f from V to an Abelian group \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{G}}$$\end{document} of order n such that the weight \documentclass[12pt]{minimal}
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\begin{document}$${w(x) = \sum_{y\in N_G(x)}f(y)}$$\end{document} of every vertex \documentclass[12pt]{minimal}
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\begin{document}$${x \in V}$$\end{document} is equal to the same element \documentclass[12pt]{minimal}
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\begin{document}$${\mu \in \mathcal{G}}$$\end{document} , called the magic constant. In this paper we will show that if G is a graph of order n = 2p(2k + 1) for some natural numbers p, k such that \documentclass[12pt]{minimal}
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\begin{document}$${\deg(v)\equiv c \mod {2^{p+1}}}$$\end{document} for some constant c for any \documentclass[12pt]{minimal}
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\begin{document}$${v \in V(G)}$$\end{document} , then there exists a \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{G}}$$\end{document} -distance magic labeling for any Abelian group \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{G}}$$\end{document} of order 4n for the composition G[C4]. Moreover we prove that if \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{G}}$$\end{document} is an arbitrary Abelian group of order 4n such that \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{G} \cong \mathbb{Z}_2 \times\mathbb{Z}_2 \times \mathcal{A}}$$\end{document} for some Abelian group \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{A}}$$\end{document} of order n, then there exists a \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{G}}$$\end{document} -distance magic labeling for any graph G[C4], where G is a graph of order n and n is an arbitrary natural number.
