  For a given group $G$ the orientably regular maps with orientation-preserving
automorphism group $G$ are used as the vertices of a graph $\O(G)$, with
undirected and directed edges showing the effect of duality and hole operations
on these maps. Some examples of these graphs are given, including several for
small Hurwitz groups. For some $G$, such as the affine groups ${\rm
AGL}_1(2^e)$, the graph $\O(G)$ is connected, whereas for some other infinite
families, such as the alternating and symmetric groups, the number of connected
components is unbounded.
