  Eigenstates of a non-Hermitian system exist on complex Riemannian manifolds,
with multiple sheets connecting at branch cuts and exceptional points (EPs).
These eigenstates can evolve across different sheets, a process that naturally
corresponds to state permutation. Here, we report the first experimental
realization of non-Abelian permutations in a three-state non- Hermitian system.
Our approach relies on the stroboscopic encircling of two different exceptional
arcs (EAs), which are smooth trajectories of order-2 EPs appearing from the
coalescence of two adjacent states. The non-Abelian characteristics are
confirmed by encircling the EAs in opposite sequences. A total of five
non-trivial permutations are experimentally realized, which together comprise a
non-Abelian group. Our approach provides a reliable way of investigating
non-Abelian state permutations and the related exotic winding effects in non-
Hermitian systems.
