  We study the three-dimensional generalized six-state clock model at values of
the energy parameters, at which the system is considered to have the same
behavior as the stacked triangular antiferromagnetic Ising model and the
three-state antiferromagnetic Potts model. First, we investigate ordered phases
by using the Monte Carlo twist method (MCTM). We confirmed the existence of an
incompletely ordered phase (IOP1) at intermediate temperature, besides the
completely ordered phase (COP) at low-temperature. In this intermediate phase,
two neighboring states of the six-state model mix, while one of them is
selected in the low temperature phase. We examine the fluctuation the mixing
rate of the two states in IOP1 and clarify that the mixing rate is very stable
around 1:1.
  The high temperature phase transition is investigated by using
non-equilibrium relaxation method (NERM). We estimate the critical exponents
beta=0.34(1) and nu=0.66(4). These values are consistent with the 3D-XY
universality class. The low temperature phase transition is found to be of
first-order by using MCTM and the finite-size-scaling analysis.
