Table 1 Some permutation symmetry groups that have been considered as examples of symmetries of coupled oscillator networks

Full permutation

\(S_{N}\)

Global or all-to-all coupling [118, 120]

Undirected ring

\(\mathbb {D}_{N}\)

Dihedral symmetry [118, 120]

Directed ring

\(\mathbb {Z}_{N}\)

Cyclic symmetry [118, 120]

Polyhedral networks

Various

[121]

Lattice networks

\(G_{1}\times G_{2}\)

\(G_{1}\) and \(G_{2}\) could be \(\mathbb {D}_{k}\) or \(\mathbb {Z}_{k}\)

Hierarchical networks

\(G_{1} \wr G_{2}\)

\(G_{1}\) is the local symmetry, \(G_{2}\) the global symmetry, and ≀ is the wreath product [122]