  We study the dark matter from an inert doublet and a complex scalar singlet
stabilized by $\mathbb{Z}_N$ symmetries. This field content is the minimal one
that allows dimensionless semi-annihilation couplings for $N > 2$. We consider
explicitly the $\mathbb{Z}_3$ and $\mathbb{Z}_4$ cases and take into account
constraints from perturbativity, unitarity, vacuum stability, necessity for the
electroweak $\mathbb{Z}_N$ preserving vacuum to be the global minimum,
electroweak precision tests, upper limits from direct detection and properties
of the Higgs boson. Co-annihilation and semi-annihilation of dark sector
particles as well as dark matter conversion significantly modify the cosmic
abundance and direct detection phenomenology.
