The group of automorphisms of the 32-vertex Dyck graph is identified as the tetrakisoctahedral group; O-4. This group has 96 elements and conserves orientation on the standard embedding of the Dyck graph on a surface of genus 3, consisting of 12 octagons. An alternative regular map of the Dyck graph on a torus is found, which is made up of 16 hexagons. Orientation on this surface is conserved by another group of 96 elements, T-4(h), which is non-isomorphic to O-4. The subgroup structures of O-4 and T-4(h) are derived, and character tables of O-4 and some of its subgroups are constructed. The symmetry representations of the Dyck graph and its topological dual are determined. Finally a molecular realization of the Dyck graph on the genus-3 `Plumber's nightmare' is proposed, which can be considered as a new type of octagonal carbon network.