To check one of my results, I need the following special example.
Is there a finite group $G$ and a generating set $S=S^{-1}$ such that the degree of Cayley graph $\Gamma(G,S)$ is at least 3 and also it is planar and edge-transitive?
Thanks so much for your helps
Parcly Taxel, kindly presented an example of a Cayley graph of an abelain group with the above properties. This example solves my problem, but I am curious to know that is there a non-abelain example too? Again thanks so much dear Parcly Taxel
Consider the abelian group $Z_2^3$ generated by the non-trivial elements of each $Z_2$ factor. The Cayley graph thus formed is a cube, which has degree 3, is edge-transitive and is planar. This gives a positive answer to the question.