The process $psi(3686) to pi^0 h_c, h_c to gamma eta_c$ has been studied with a data sample of $106 pm 4$ million $psi(3686)$ events collected with the BESIII detector at the BEPCII storage ring. The mass and width of the $P$-wave charmonium spin-singlet state $h_c(^1P_1)$ are determined by simultaneously fitting distributions of the $pi^0$ recoil mass for 16 exclusive $eta_c$ decay modes. The results, $M(hc) = 3525.31 pm 0.11 {rm (stat.)} pm 0.14 {rm (syst.)}$,MeV/$c^2$ and $Gamma(hc) = 0.70 pm 0.28 pm 0.22$,MeV, are consistent with and more precise than previous measurements. We also determine the branching ratios for the 16 exclusive $eta_c$ decay modes, five of which have not been measured previously. New measurements of the $eta_c$ line-shape parameters in the $E1$ transition $h_ctogammaeta_c$ are made by selecting candidates in the $h_c$ signal sample and simultaneously fitting the hadronic mass spectra for the 16 $eta_c$ decay channels. The resulting $eta_c$ mass and width values are $M(eta_c) = 2984.49 pm 1.16 pm 0.52$,MeV/$c^2$ and $Gamma(eta_c) = 36.4 pm 3.2 pm 1.7$,MeV.
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