  The strong and radiative decays of the low-lying $S$- and $P$-wave
$\Lambda_{c(b)}$, $\Sigma_{c(b)}$, $\Xi_{c(b)}$, $\Xi_{c(b)}'$ and
$\Omega_{c(b)}$ baryons are systematically studied in a constituent quark
model. We find that the radiative decay mode $\Lambda_b^0\gamma$ could be very
useful for us to establish the missing neutral states $\Sigma_b^{0}$ and
$\Sigma_b^{*0}$. Our calculation shows that most of those missing
$\lambda$-mode $P$-wave singly heavy baryons have relatively narrow decay width
of less than 30 MeV. Their dominant strong and radiative decay channels can be
ideal for searching for their signals in future experiment. Configuration
mixings between the $J^P=1/2^-$ states $|^2P_{\lambda} \frac{1}{2}^- \rangle$
and $|^4P_{\lambda} \frac{1}{2}^- \rangle$ of the $\mathbf{6}_F$ multiplet are
discussed. The $\Xi_c(2930)^0$ state seems to favor the mixed state $|\Xi_c'~
P_{\lambda}\frac{1}{2}^-\rangle_2$ with $J^P=1/2^-$. The $\Sigma_c(2800)$
resonance may be assigned to $|\Sigma_c~ ^2P_{\lambda} \frac{3}{2}^- \rangle$
with $J^P=3/2^-$ or $|\Sigma_c~ ^4P_{\lambda} \frac{5}{2}^- \rangle$ with
$J^P=5/2^-$. In general, the excitations of $|^2P_{\lambda} \frac{3}{2}^-
\rangle$ and $|^4P_{\lambda} \frac{5}{2}^- \rangle$ of the $\mathbf{6}_F$
multiplet have similar strong decay properties. In order to to identify them,
angular distributions of their decays in either strong decay modes or radiative
transitions should be needed.
