The automorphism group of the Dyck graph is a group of order 192.[4] It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore, the Dyck graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the Foster census, the Dyck graph, referenced as F32A, is the only cubic symmetric graph on 32 vertices.[5]
^Ceulemans, A. (2004), "The tetrakisoctahedral group of the Dyck graph and its molecular realization.", Molecular Physics, 102 (11): 1149–1163, doi:10.1080/00268970410001728780, S2CID97973403.