Distinguishing index of maps

The distinguishing number of a group $A$ acting on a finite set $\Omega$, denoted by $D(A,\Omega )$, is the least $k$ such that there is a $k$-coloring of $\Omega$ which is preserved only by elements of $A$ fixing all points in $\Omega$. For a map $M$, also called a cellular graph embedding or ribbon graph, the action of $Aut\left(M\right)$ on the vertex set $V$ gives the distinguishing number $D\left(M\right)$. It is known that $D\left(M\right)\le 2$ whenever $\left|V\right|>10$. The action of $Aut\left(M\right)$ on the edge set $E$ gives the distinguishing index $D^{\prime }\left(M\right)$, which has not been studied before. It is shown that the only maps $M$ with $D^{\prime }\left(M\right)>2$ are the following: the tetrahedron; the maps in the sphere with underlying graphs $C_n$, or $K_{1,n}$ for $n=3,4,5$; a map in the projective plane with underlying graph $C_4$; two one-vertex maps with 4 or 5 edges; one two-vertex map with 4 edges; or any map obtained from these maps using duality or Petrie duality. There are 39 maps in all.