  Periodic and quasi-periodic orbits of the $n$-body problem are critical
points of the action functional constrained to the Sobolev space of symmetric
loops. Variational methods yield collisionless orbits provided the group of
symmetries fulfills certain conditions (such as the \emph{rotating circle
property}). Here we generalize such conditions to more general group types and
show how to constructively classify all groups satisfying such hypothesis, by a
decomposition into irreducible transitive components. As examples we show
approximate trajectories of some of the resulting symmetric minimizers.
