Search for the $X\left(2370\right)$ and observation of ${\eta }_{c}\to \eta \eta \eta \prime $ in $J/\psi \to \gamma \eta \eta \eta \prime $

Using a sample of $1.31×{10}^{9}\text{}\text{}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 $X\left(2370\right)$ and ${\eta }_{c}$ in the $\eta \eta {\eta }^{\prime }$ invariant mass distribution. No significant signal for the $X\left(2370\right)$ is observed, and we set an upper limit for the product branching fraction of $B\left(J/\psi \to \gamma X\left(2370\right)·B\left(X\left(2370\right)\to \eta \eta {\eta }^{\prime }\right)<9.2×{10}^{-6}$ at the 90% confidence level. A clear ${\eta }_{c}$ signal is observed for the first time, yielding a product branching fraction of $B\left(J/\psi \to \gamma {\eta }_{c}\right)·B\left({\eta }_{c}\to \eta \eta {\eta }^{\prime }\right)=\phantom{\rule{0ex}{0ex}}\left(4.86±0.62\left(\mathrm{stat}\right)±0.45\left(\mathrm{sys}\right)\right)×{10}^{-5}$.