  Using Monte Carlo simulations and finite-size scaling analysis, we show that
the $q$-state clock model with $q=6$ on the simple cubic lattice with open
surfaces has a rich phase diagram; in particular, it has an extraordinary-log
phase, besides the ordinary and extraordinary transitions at the bulk critical
point. We prove numerically that the presence of the intermediate
extraordinary-log phase is due to the emergence of an O(2) symmetry in the
surface state before the surface enters the $Z_{q}$ symmetry-breaking region as
the surface coupling is increased at the bulk critical point, while O(2)
symmetry emerges for the bulk. The critical behaviors of the extraordinary-log
transition, as well as the ordinary and the special transition separating the
ordinary and the extraordinary-log transition are obtained.
