Example where automorphism group of Cayley graph of $G$ is not $G$

Question

Clearly, $G$ is a subgroup of the automorphism group of a Cayley graph $\Gamma$ of $G$, because $G$ acts on $\Gamma$. I know plenty of examples where $G = \text{Aut}(\Gamma)$, but what is an example where the automorphism group of $\Gamma$ is larger than $G$?

Answer 1

Assuming that $\Gamma$ is the undirected graph underlying the Cayley graph*, easy examples are found with cyclic groups:

• The Cayley graph of $\mathbb{Z}_n=\langle 1\rangle$ is a regular $n$-gon. The automorphism group of a regular $n$-gon is the dihedral group of order $2n$, $D_{n}$.

• The Cayley graph of $\mathbb{Z}=\langle 1\rangle$ is a straight line. The automorphism group of a straight line is the infinite dihedral group, $D_{\infty}$.

In each of these cases, $G$ has index two in $\operatorname{Aut}(\Gamma)$. The extra symmetries come from "flipping" the graph.

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*Without this assumption, $G\cong\operatorname{Aut}(\Gamma)$. That is, a group $G$ is the automorphism group of any associated Cayley graph. Which is kinda the point of Cayley graphs...