Abstract
The group of automorphisms of the 32-vertex Dyck graph is identified as the tetrakisoctahedral group, 4 O. 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, 4 Th , which is non-isomorphic to 4 O. The subgroup structures of 4 O and 4 Th are derived, and character tables of 4 O 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.
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