Classifying vertex-transitive dihedral graphs

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Is there some way to generate adjacency matrices for all graphs with $n$ vertices that have a dihedral $D_{n}$ symmetry which acts transitively on the vertices? (the vertex set cardinality $n$ being the same as the $n$ in the $D_{n}$ subscript) For example for circulant graphs, each row of the adjacency matrix will be a cyclic permutation of the previous row. Is there some more general procedure?