A Cayley diagram is a choice, not a fact about the group. Another drawing of this group — in a book, or on a different generating set — may look nothing like this one.
Which vertices are joined, and by which generator, is what the group fixes. Where they sit is this drawing's choice, made to show as much of the group's symmetry as the plane allows.
Change the generating set and the graph itself changes, not only its picture: C6 is a hexagon on one generator, and a triangular prism on its elements of order 3 and 2.
Even a minimal generating set is not unique. C4 × C2 has three edges at each vertex on { a, b } and four on { a, ab }, and both sets are minimal.
Finite Group Databasebeta