  For the $Z_{3}$-symmetric lattice QCD-like theory ($Z_3$-QCD), in which
$SU(3)$ gauge theory is coupled with three fundamental Wilson quarks with
flavor-dependent twisted boundary conditions, we calculate the expectation
values of Polyakov loop and chiral condensate as functions of temperature on
$16^3 \times4$ and $20^3 \times 4$ lattices with $m_{PS}/m_{V}=0.70$ fixed. We
find the first-order phase transition with respect to the $Z_{3}$ center
symmetry, where the Polyakov loop exhibits a hysteresis depending on the
initial condition of thermalization process. We also show that the crossover
behavior of chiral condensate around the critical temperature of the center
transition and the manifestation of flavor symmetry breaking in the
high-temperature phase.
