  We present the ground and excited state spectra of doubly and triply-charmed
baryons by using lattice QCD with dynamical clover fermions. A large set of
baryonic operators that respect the symmetries of the lattice and are obtained
after subduction from their continuum analogues are utilized. Using novel
computational techniques correlation functions of these operators are generated
and the variational method is exploited to extract excited states. The lattice
spectra that we obtain have baryonic states with well-defined total spins up to
7/2 and the low lying states remarkably resemble the expectations of quantum
numbers from SU(6) $\otimes$ O(3) symmetry. Various energy splittings between
the extracted states, including splittings due to hyperfine as well as
spin-orbit coupling, are considered and those are also compared against similar
energy splittings at other quark masses. Using those splittings for
doubly-charmed baryons, and taking input of experimental $B_c$ meson mass, we
predict the mass splittings of $B^*_c - B_c$ to be about 80 $\pm$ 8 MeV and
$m_{\Omega_{ccb}} = 8050\pm10$ MeV.
