  Let $G$ be a finite group acting on a closed manifold $M$. We estimate the
size of a minimal triangulation of $M$ for which the action of $G$ is
simplicial and regular. We show that the number of vertices of such
triangulations are bounded bellow by the \emph{$G$-covering type} of $M$,
defined as the minimal cardinality of a $G$-equivariant good cover of a space
that is $G$-homotopy equivalent to $M$. The $G$-covering type is a $G$-homotopy
invariant, so it can be estimated by other $G$-invariants like the equivariant
LS-category, $G$-genus and the multiplicative structure of any equivariant
cohomology theory. Among many examples, we compute the $G$-covering type of
regular $G$-graphs, give a complete description of the number of vertices and
their orbits for orientation-preserving finite group actions on surfaces and
estimate the $G$-covering type of the projectivizations of complex
representations of $G$.
