The strong and radiative decays of the low-lying $lambda$-mode $D$-wave $Lambda_{c(b)}$, $Sigma_{c(b)}$, $Xi_{c(b)}$, $Xi_{c(b)}$, and $Omega_{c(b)}$ baryons are studied in a constituent quark model. Our calculation shows the following: (i) The missing $lambda$-mode $D$-wave $Omega_{c(b)}$, $Lambda_{b}$, and $Xi_{b}$ baryons have a relatively narrow decay width of a few MeV or a few tens of MeV and their dominant strong and radiative decay channels can be ideal for searching for their signals in future experiments. (ii) The $lambda$-mode $1D$-wave excitations in the $Sigma_{c(b)}$ and $Xi_{c(b)}$ families appear to have a relatively broad width of $sim 50-200$ MeV.Most of the $1D$-wave states have large decay rates into the $1P$-wave heavy baryons via the pionic or kaonic strong decay processes, which should be taken seriously in future observations. (iii) Both $Lambda_c(2860)$ and $Xi_c(3050)$ seem to favor the $J^P=3/2^+$ excitation $|^2D_{lambdalambda} frac{3}{2}^+ rangle$ of $bar{mathbf{3}}_F$, while both $Lambda_c(2880)$ and $Xi_c(3080)$ may be assigned as the $J^P=5/2^+$ excitation $|^2D_{lambdalambda} frac{5}{2}^+ rangle$ of $bar{mathbf{3}}_F$. The nature of $Xi_c(3050)$ and $Xi_c(3080)$ could be tested by the radiative transitions $Xi_c(3055)^0to Xi_c(2790)^0 gamma$ and $Xi_c(3080)^0 to Xi_c(2815)^0 gamma$, respectively.
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