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I'm looking for graph $G$, $V(G) >2$, such that$G \cong Cay(Aut(G), S) $, where $Cay$ is the Cayley graph (considered as undirected graph) and $S$ is minimal generator of $G$.

I've written a GAP code that looped through $\Gamma$ groups, and formed $G'=Cay(\Gamma)$ and $G''=Cay(Aut(G')) $. After it checked if $G' \cong G''$ is true. It happened to be false for the first 1000 groups, except when $\Gamma$ is the trivial group and when it is isomorphic to $C_2$.

Now, I am not even sure what generator set was used by GAP. Is it possible that for other generator set it would work? Maybe if it isn't minimal?

Thanks in advance!

Máté Kadlicskó
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  • See https://math.stackexchange.com/questions/1098115/when-is-the-automorphism-group-of-the-cayley-graph-of-g-just-g – Misha Lavrov Nov 05 '18 at 23:45
  • I'm guessing you mean $Aut(Cay(..$ not $Cay(Aut(...$. – verret Nov 05 '18 at 23:57
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    If you want a specific example, take $G$ to be the dihedral group of order $12$, let $a$ be a basic rotation (of order 6) and $b$ a reflection, then take $S={a,a^{-1},b,ba,ba^3}$. The automorphism group of this graph is $G$. (There are no examples of order less than $12$.) – verret Nov 06 '18 at 00:04

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