# A question about groups, that are isomorphic to the automorphism groups of their cycle graphs

Suppose $$G$$ is a finite group. Let’s call an element $$g \in G$$ primitive, iff $$\forall h \in G$$, such that $$\exists n \in \mathbb{N}, h^n = g$$ it is true, that $$\langle h \rangle = \langle g \rangle$$. Suppose $$P(G)$$ is the set of all primitive elements of $$G$$. It is not hard to see, that $$\langle P(G) \rangle = G$$ and moreover $$\forall g \in G \exists h \in P(G) g \in \langle h \rangle$$.

Now, suppose the cycle graph of $$G$$ is an undirected graph $$Cyc(G) = (V, E)$$, such that the set of all vertices $$V = G$$ and the set of all edges is $$E = \{(g^n, g^{n+1})|g \in P(G) n \in \mathbb{N}\}$$.

Now, let’s consider automorphism groups of cycle graphs. I have managed to find them for the following groups: $$Aut(Cyc(E)) \cong E$$ $$Aut(Cyc(C_2)) \cong C_2$$ $$Aut(Cyc(C_2^n)) \cong S_{2^n - 1}, \forall n > 1$$ $$Aut(Cyc(C_m)) \cong D_m, \forall m > 2$$ $$Aut(Cyc(C_m^n)) \cong C_2^{\frac{m^n - 1}{m - 1}} \times S_{\frac{m^n - 1}{m - 1}}, \forall m > 2, n > 1$$ $$Aut(Cyc(D_m)) \cong C_2 \times S_m, \forall m > 2$$ $$Aut(Cyc(Q_8)) \cong C_2 \times S_6$$

The other finite groups seem to have too complicated cycle graphs to find all their automorphisms manually.

As you may have noticed, among the groups above, only $$E$$ and $$C_2$$ satisfy the condition $$G \cong Aut(Cyc(G))$$. The question is - are there any other such finite groups?

This question can also be reworded as: «Does $$G \cong Aut(Cyc(G))$$ imply $$|G| \leq 2$$

If a counterexample exist, it is not a direct product of isomorphic cyclic groups, not a dihedral group and not the quaternion group. However, I do not know, how to deal here with groups, that do not fall into any of those classes.