  Higgs doublets may come in three generations. The scalar sector of the
resulting three-Higgs-doublet model (3HDM) may be constrained by global
symmetry groups $G$ leading to characteristic phenomenology. There exists the
full list of symmetry groups $G$ realizable in the 3HDM scalar sector and the
expressions for $G$-symmetric scalar potentials written in special bases where
the generators of $G$ take simple form. However recognizing the presence of a
symmetry in a generic basis remains a major technical challenge, which impedes
efficient exploration of the 3HDM parameter space. In this paper, we solve this
problem using the recently proposed approach, in which basis-independent
conditions are formulated as relations among basis-covariant objects. We
develop the formalism and derive basis-independent necessary and sufficient
conditions for the 3HDM scalar sector to be invariant under each of the
realizable symmetry group. We also comment on phenomenological consequences of
these results.
