  We consider a mixture of one-dimensional strongly interacting Fermi gases up
to six components, subjected to a longitudinal harmonic confinement. In the
limit of infinitely strong repulsions we provide an exact solution which
generalizes the one for the two-component mixture. We show that an imbalanced
mixture under harmonic confinement displays partial spatial separation among
the components, with a structure which depends on the relative population of
the various components. Furthermore, we provide a symmetry characterization of
the ground and excited states of the mixture introducing and evaluating a
suitable operator, namely the conjugacy class sum. We show that, even under
external confinement, the gas has a definite symmetry which corresponds to the
most symmetric one compatible with the imbalance among the components. This
generalizes the predictions of the Lieb-Mattis theorem for a fermionic mixture
with more than two components.
