It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser G_e of every edge e is dihedral of order 4 and the stabiliser G_#upsilon# of each vertex #upsilon# is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incident with #upsilon#. Such a regular embedding can be realised on an orientable surface if and only if the group G has a subgroup H of index 2 such that H_#upsilon# is the cyclic subgroup of index 2 in G_#upsilon#. An analogous result is proved for orientably-regular embeddings.
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