  We investigate QCD-like theory with exact center symmetry, with emphasis on
the finite-temperature phase transition concerning center and chiral
symmetries. On the lattice, we formulate center symmetric $SU(3)$ gauge theory
with three fundamental Wilson quarks by twisting quark boundary conditions in a
compact direction ($Z_3$-QCD model). We calculate the expectation value of
Polyakov loop and the chiral condensate as a function of temperature on 16^3 x
4 and 20^3 x 4 lattices along the line of constant physics realizing
$m_{PS}/m_{V}=0.70$. We find out the first-order center phase transition, where
the hysteresis of the magnitude of Polyakov loop exists depending on
thermalization processes. We show that chiral condensate decreases around the
critical temperature in a similar way to that of the standard three-flavor QCD,
as it has the hysteresis in the same range as that of Polyakov loop. We also
show that the flavor symmetry breaking due to the twisted boundary condition
gets qualitatively manifest in the high-temperature phase. These results are
consistent with the predictions based on the chiral effective model in the
literature. Our approach could provide novel insights to the nonperturbative
connection between the center and chiral properties.
