In the literature, one can find at least three different genus parameters associated with a finite group: genus, symmetric genus, and strong symmetric genus. While the genus of a group is defined in terms of the (undirected, non-edge-colored) Cayley graphs, plain graphs are not adequate for modeling symmetric and strongly symmetric embeddings and thus cannot be used directly for determining the symmetric and strong symmetric genus of a group. We propose (generalized) action graphs to model such embeddings. Although action graphs are a wider class of edge-colored, partially directed graphs than Cayley color (di)graphs, the idea of symmetric and strongly symmetric embedding can be applied to them. In addition, we present some applications of action graphs. 

This is joint work with Thomas W. Tucker. 

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