Let $X$ be a connected graph on $2^n$ vertices for $n ≥ 1$. Prove that $X$ is a Cayley graph of $\mathit{Z}\,_n^2$ if and only if X has a $1$-factorization such that the union of any two $1$-factors consists of the disjoint union of $4$-cycles.
I know that $X(G,C)$ is called a Cayley graph with its property. In this problem, What is C?