  In this paper, we study the feeble sterile neutrino portal dark matter under
the $Z_3$ symmetry. The dark sector consists of one fermion singlet $\chi$ and
one scalar singlet $\chi$, which transforms as $\chi\to e^{i2\pi/3}\chi,
\phi\to e^{i2\pi/3}\phi$ under the $Z_3$ symmetry. Regarding fermion singlet
$\chi$ as the dark matter candidate, the new interaction terms $y_\chi \phi
\bar{\chi^c}\chi$ and $\mu\phi^3/2$ could induce various new production
channels. For instance, when $m_\phi>2m_\chi$, the pair decay $\phi\to\chi\chi$
could be the dominant channel, rather than the delayed decay $\phi\to\chi\nu$.
Another appealing scenario is when the dark sector is initially produced
through the scattering process as $NN\to\chi\chi,
NN\to\phi\phi,h\nu\to\chi\phi$, then the semi-production processes $N
\chi\to\phi\phi, N\phi\to\phi\chi, N\chi\to\chi\chi$ could lead to the
exponential growth of dark sector abundances. The phenomenology of sterile
neutrino and the cosmological impact of the dark scalar are also considered in
the $Z_3$ symmetric model.
