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The problem of finding minimal vertex number of graphs with a given automorphism group is addressed in this article for the case of cyclic groups. This problem was considered earlier by other authors. We give a construction of an undirected graph having $4n+6$ vertices and automorphism group cyclic of order $4n$, $n\ge 1$. As a special case we get graphs with $2^k+6$ vertices and cyclic automorphism groups of order $2^k$. It can revive interest in related problems.