Non-Abelian permutations meet D4 group in a four-state non-Hermitian system

Encircling non-Hermitian exceptional points (EPs) leads to the swapping of eigenstates, which naturally corresponds to the different state permutations. Here, we find that non-Abelian permutations meet the $D_4$ group in a four-state non-Hermitian system. By encircling three different exceptional arcs within a three-dimensional parameter space, we present seven eigenvalue permutations and correlate them with seven nontrivial operations in the non-Abelian $D_4$ group. Furthermore, our theory reveals the topological properties of higher-order EPs, improves the understanding of EPs in non-Hermitian systems, and offers a robust methodology for examining non-Abelian properties in systems with multiple EPs.