  We consider a complex scalar field as SIMP dark matter in models with gauged
$Z_3$ discrete symmetry appearing as a remnant of dark local $U(1)$. Dark
matter (DM) annihilates dominantly by the 3-to-2 scattering, due to the DM
cubic coupling in combination with the DM quartic coupling or the $Z'$ gauge
and dark Higgs couplings. We show that a light $Z'$ gauge boson makes DM in
kinetic equilibrium with thermal plasma at freeze-out and it affects the DM
relic density and perturbativity/unitarity constraints for DM
self-interactions. We show that the large DM self-interactions are consistent
with solving small-scale structure problems and explaining the DM halo
separation recently observed in Abell 3827 cluster. Various bounds on the model
from the SIMP conditions, DM self-interactions, $Z'$ searches, DM
direct/indirect detection experiments, and Higgs signals, are also discussed.
