  Space group symmetries dictate the energy degeneracy of quasiparticles (e.g.,
electronic, photonic) in crystalline structures. For spinless systems, there
can only be double or triple degeneracies protected by these symmetries, while
other degeneracies are usually taken as \textit{accidental}. In this Letter we
show that it is possible to design higher degeneracies exploring site
permutation symmetries. These design principles are shown to be satisfied in
previously studied lattices, and new structures are proposed with three, four
and five degeneracy points for spinless systems. The results are general and
apply to different quasiparticle models. Here, we focus on a tight-binding
approach for the electronic case as a proof of principle. The resulting
high-degeneracy points are protected by the site-permutation symmetries,
yielding pseudospin-1 and -2 Dirac fermions. The strategy proposed here can be
used to design lattices with high-degeneracy points in electronic (e.g.
metal-organic frameworks), photonic, phononic, magnonic and cold-atom systems.
