  Using a sample of $1.31\times10^{9} ~J/\psi$ events collected with the BESIII
detector, we perform a study of $J/\psi\to\gamma\eta\eta\eta^{\prime}$ to
search for the $X(2370)$ and $\eta_{c}$ in the $\eta\eta\eta^{\prime}$
invariant mass distribution. No significant signal for the $X(2370)$ is
observed, and we set an upper limit for the product branching fraction of
${\cal B}(J/\psi\to\gamma X(2370)\cdot{\cal B}(X(2370)\to\eta\eta\eta^{\prime})
< 9.2\times10^{-6}$ at the 90% confidence level. A clear $\eta_{c}$ signal is
observed for the first time, yielding a product branching fraction of ${\cal
B}(J/\psi\to \gamma\eta_{c})\cdot{\cal B}(\eta_{c}\to \eta\eta\eta^{\prime}) =
(4.86\pm0.62~({\rm stat.})\pm0.45~({\rm sys.}))\times10^{-5}$.
