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The existence of a regular, self-dual and self-Petrie-dual map of any given even valency has been proved by D. Archdeacon, M. Conder and J. Siraň (2014). In this paper we extend this result to any odd valency ≥ 5 . This is done using algebraic number theory and maps defined on the groups PSL(2,  p ) in the case of odd prime valency ≥ 5 and valency 9 , and using coverings for the remaining odd valencies.

Regular self-dual and self-Petrie-dual maps of arbitrary valency