We study a linear description of $PC^k_n$ the convex hull of incidence vectors of all the cycles consisting of exactly $k$ edges ($k \ge 4$) in $K_n$, the complete undirected graph with $n$ vertices. First, we describe some properties of $PC^k_n$. Second, we discuss relations between $PC^k_n$ and $PC^{k'}_{n'}$ with $k' > k$ and $n' > n$. Then we give three lifting algorithms that transform a facet of $PC^k_n$ into facets of $PC^k_{n'}$ with $n' > n$. Finally, we provide an integer formulation and a partial linear description of $PC^k_n$.