  In the framework of the heavy baryon chiral perturbation theory (HBChPT), we
calculate the radiative decay amplitudes of the singly heavy baryons up to the
next-to-next-to-leading order (NNLO). In the numerical analysis, we adopt the
heavy quark symmetry to relate some low energy constants (LECs) with those LECs
in the calculation of the magnetic moments. We use the results from the lattice
QCD simulation as input. With a set of unified LECs, we obtain the numerical
(transition) magnetic moments and radiative decay widths. We give the numerical
results for the spin-$1\over 2$ sextet to the spin-$1\over 2$ antitriplet up to
the next-to-leading order (NLO). The nonvanishing $\Gamma(\Xi_c^{'0}
\rightarrow \Xi^0_c\gamma)$ and $\Gamma(\Xi_c^{*0} \rightarrow \Xi^0_c \gamma)$
solely arise from the U-spin symmetry breaking, and do not depend on the
lattice QCD inputs up to NLO. We also systematically give the numerical
analysis of the magnetic moments of the spin-$1\over 2$, spin-$3\over 2$ sextet
and their radiative decay widths up to NNLO. In the heavy quark limit, the
radiative decays between the sextet states happen through the magnetic dipole
(M1) transitions, while the electric quadrupole (E2) transition does not
contribute. We also extend the same analysis to the single bottom baryons.
