We discuss a new family of cubic graphs, which we call $SGP$-graphs, that bears a close resemblance to the family of generalized Petersen graphs; both in definition and properties. The focus of our paper is on determining the algebraic properties of graphs from our new family. We look for highly symmetric graphs, e.g., graphs with large automorphism groups, vertex- or arc-transitive graphs. In particular, we present arithmetic conditions for the defining parameters that guarantee that graphs with these parameters are vertex-transitive or Cayley, and we find one arc-transitive $SGP$-graph which is neither a $CQ$ graph of Feng and Wang, nor a generalized Petersen graph. 

Joint work with Katarina Jasencakova and Robert Jajcay.