Constructing an isomorphism from dihedral group $D_5$ to Grötzsch graph automorphism group

Question

It is known that the Diedral group $D_5$ and the Grötzsch graph automorphism group are isomorphic. Could someboby provide me and explicit construction of such an isomorphism?

Answer 1

The graph has one vertex of degree 5, which I'll call the central vertex, 5 vertices of degree 3, which I'll call the inner ring, and 5 vertices of degree 4, which I'll call the outer ring. Each vertex in the inner ring is adjacent to the central vertex, so it's clear that any automorphism must leave the central vertex fixed, and permute the vertices of the inner ring. Furthermore, each vertex in the outer ring is adjacent to exactly two vertices in the inner ring, so the permutation on the inner determines the automorphism.

So you just have to show that the only admissible permutations of the inner that extend properly to the outer ring, are those of the dihedral group.