Topological phases characterized by non-Abelian charges have garnered increasing attention recently. Although Floquet (periodic-driving) higher-order topological phases have been explored at the single-particle level, the role of couplings in non-Abelian topological insulators with multiple entangled energy gaps remains incompletely understood. In this work, we extend previous research by investigating higher-order topological phases featuring non-Abelian charges through Floquet engineering. Here we construct a model for two-dimensional non-Abelian higher-order topological phases on a square lattice subjected to two-step periodic driving. We find that the corner and edge states emerge and appear in all energy gaps despite the quaternion charge being trivial ($q=1$). Moreover, spatially exchanging the driving generates exotic interface modes—a hallmark of non-Abelian dynamics, namely, noncommutativity. Notably, although the composite Chern number is trivial due to $\mathcal{PT}$ symmetry, the Floquet system hosts nontrivial higher-order and interface topology characterized by the Stiefel Whitney invariant $w_2$ and gap-resolved Zak phase mismatches. We further reveal that the configuration of these quaternion-charge edge states is entirely determined by the quadruple degenerate phase-band singularities in the time evolution. Our work provides a platform for studying higher-order topological states and nonequilibrium quantum dynamics.

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• Topological insulators
• Topological phase transition
• Topological phases of matter
• Floquet systems