Rubik's cube question showing surjectivity

Question

Suppose $f: G \rightarrow S_C$ be the action of Rubik's group G on the collection $C$ of corner cubelets.

I want to show that the action $f$ is onto $S_C$.

Answer 1

Once you know that there is a move that interchanges two neighboring corners and leaves the other 6 corners where they were, show that $f(G)$ contains a set of generators for $S_C$.

(Hint: You can list all of the corner positions in an order such that each position is physically a neighbor of the one that comes after it).