  The set of all centralizers of elements in a finite group $G$ is denoted by
$Cent(G)$ and $G$ is called $n-$centralizer if $|Cent(G)| = n$. In this paper,
the structure of centralizers in a non-abelian finite group $G$ with this
property that $\frac{G}{Z(G)} \cong Z_{p^2} \rtimes Z_{p^2}$ is obtained. As a
consequence, it is proved that such a group has exactly $[(p+1)^2+1]$ element
centralizers and the structure of the commuting conjugacy class graph of $G$ is
completely determined.
