  We explore the phase structure and symmetry breaking in four-dimensional
SU(3) gauge theory with one spatial compact dimension on the lattice ($16^3
\times 4$ lattice) in the presence of fermions in the adjoint representation
with periodic boundary conditions. We estimate numerically the density plots of
the Polyakov loop eigenvalues phases, which reflect the location of minima of
the effective potential in the Hosotani mechanism. We find strong indication
that the four phases found on the lattice correspond to SU(3)-confined,
SU(3)-deconfined, SU(2) x U(1), and U(1) x U(1) phases predicted by the
one-loop perturbative calculation. The case with fermions in the fundamental
representation with general boundary conditions, equivalent to the case of
imaginary chemical potentials, is also found to support the $Z_3$ symmetry
breaking in the effective potential analysis.
