  In this paper, we consider the sterile neutrino portal dark matter with $Z_3$
symmetry. This model further extends the canonical type-I seesaw with a fermion
singlet $\chi$ and a scalar singlet $\phi$. Under the $Z_3$ symmetry, the dark
sector transforms as $\chi\to e^{i2\pi/3}\chi, \phi\to e^{i2\pi/3}\phi$, while
the standard model particles and the sterile neutrino $N$ transform trivially.
Besides the interactions as $y_{N} \phi \bar{\chi}N$ and $\lambda_{H\phi}
(H^\dag H) (\phi^\dag \phi)$ allowed in the $Z_2$ symmetry, the $Z_3$ symmetry
also introduces two new terms, i.e., $y_\chi \phi \overline{\chi^{c}} \chi$ and
$\mu\phi^3/2$. These new interactions induce additional semi-annihilation
processes as $\chi\chi\to N\chi$ and $\phi\phi\to h\phi$ for WIMP dark matter.
We then perform a comprehensive analysis of the phenomenology of this $Z_3$
symmetric model. Viable parameter space is explored under the constraints from
dark matter relic density, Higgs invisible decay, indirect and direct detection
for both fermion and scalar dark matter. We find that the semi-annihilation
channels $\chi\chi\to N\chi$ and $\phi\phi\to N\chi$ can lead to quite
different phenomena from the $Z_2$ symmetric model, which provides a viable
pathway to distinguish these two kinds of model.
