In this work, we propose dual mechanical metamaterial designs, which exhibit both non-Abelian wave packet dynamics and non-Abelian band topology. We show that the band structure of self-dual metamaterials possesses a global twofold degeneracy and three tilted Dirac cones. Theoretical derivations demonstrate that non-Abelian wave packet propagation occurs when the Dirac cones are broken by perturbing the lattice masses. Breaking the twofold degeneracy by altering the lattice angle away self-duality, we also observe non-Abelian nodal braiding, which we theoretically characterize by introducing the Euler class and frame charge. We validate the duality of band structures in our design with both numerical calculations and experiments. Our metamaterial may provide a useful platform for quantumlike computation in mechanical systems with elastic wave propagation.

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• Mechanical & acoustical properties
• Topological materials
• Band structure methods
• Group theory