  High harmonic generation (HHG) has unleashed the power of strong laser
physics in solids. Here we investigate HHG from a large system, solid C$_{60}$,
with 240 valence electrons engaging harmonic generation at each crystal
momentum, the first of this kind. We employ the density functional theory and
the time-dependent Liouville equation of the density matrix to compute HHG
signals. We find that under a moderately strong laser pulse, HHG signals reach
15th order, consistent with the experimental results from C$_{60}$ plasma. The
helicity dependence in solid C$_{60}$ is weak, due to the high symmetry. In
contrast to the general belief, HHG is unsuitable for band structure mapping in
C$_{60}$. However, we find a window of opportunity using a long wavelength,
where harmonics are generated through multiple-photon excitation. In
particular, the 5th order harmonic energies closely follow the transition
energy dispersion between the valence and conduction bands. This finding is
expected to motivate future experimental investigations.
