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The usual discrete symmetry operations in quantum mechanics i.e., C, P, and T, are representations of $\mathbb{Z}_2$. Are there examples of other discrete symmetry groups that appear in nature?

I know one answer to this is discrete rotational symmetry in crystals and quasicrystals. [Also, lattice gauge theories can have any discrete symmetry group you want, but this is not really something that appears in nature...] Other examples?

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  • $\begingroup$ This post (v2) seems like a list question. $\endgroup$
    – Qmechanic ♦
    Commented Apr 14, 2017 at 21:34
  • $\begingroup$ Flowers, starfish in biology, the eightfold way for hadrons. Systems of indistinguishable bosons display the symmetric group symmetry, and of fermions the alternating group symmetry, see permutation symmetry. $\endgroup$
    – Conifold
    Commented Apr 14, 2017 at 23:09
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    $\begingroup$ @Conifold I mean symmetries at the level of the Hamiltonian/Lagrangian. The permutation symmetry you mention is an example of that. From my cursory reading about the eightfold way, it appears that the relevant symmetry group there is SU(3), which is continuous. If I am mistaken, please feel free to elaborate about the eightfold way as an answer. $\endgroup$
    – Yly
    Commented Apr 14, 2017 at 23:17
  • $\begingroup$ Crystal symmetries do not seem to fit into a Lagrangian either, but as long a Lagrangian displays any continuous (Lie) group symmetry the solutions may display symmetries from any of its discrete subgroups, and they have plenty. Also, often the standard basis elements in a representation of a Lie group generate a discrete group, e.g. the eightfold way in the fundamental representation of SU(3) $\endgroup$
    – Conifold
    Commented Apr 14, 2017 at 23:38

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