  Given graphs $G$ and $H$, the generalized Tur\'{a}n number ${\rm ex}(G,H)$ is
the maximum number of edges in an $H$-free subgraph of $G$. In this paper, we
obtain an asymptotic upper bound on ${\rm ex}(CT_n,C_{2l})$ for any $n \ge 3$
and $l\geq2$, where $C_{2l}$ is the cycle of length $2l$ and $CT_n$ is the
complete transposition graph which is defined as the Cayley graph on the
symmetric group ${\rm S}_n$ with respect to the set of all transpositions of
${\rm S}_n$.
