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We study the three-dimensional generalized six-state clock model at values ofthe energy parameters, at which the system is considered to have the samebehavior as the stacked triangular antiferromagnetic Ising model and thethree-state antiferromagnetic Potts model. First, we investigate ordered phasesby using the Monte Carlo twist method (MCTM). We confirmed the existence of anincompletely ordered phase (IOP1) at intermediate temperature, besides thecompletely ordered phase (COP) at low-temperature. In this intermediate phase,two neighboring states of the six-state model mix, while one of them isselected in the low temperature phase. We examine the fluctuation the mixingrate of the two states in IOP1 and clarify that the mixing rate is very stablearound 1:1.  The high temperature phase transition is investigated by usingnon-equilibrium relaxation method (NERM). We estimate the critical exponentsbeta=0.34(1) and nu=0.66(4). These values are consistent with the 3D-XYuniversality class. The low temperature phase transition is found to be offirst-order by using MCTM and the finite-size-scaling analysis

Ordered phase and phase transitions in the three-dimensional generalized six-state clock model