Figure 6

The dihedral group$\mathcal{A}\mathbf{=}{\mathit{D}}_{\mathbf{4}}$is the automorphism group of the graph$\mathit{G}\mathbf{=}{\mathit{C}}_{\mathbf{4}}$. The automorphism $T_1=\mathrm{Id}$ has 4 fixed vertices of index 1 and 4 fixed edges of index −1 so that $L(T_1)=0$. There are 4 rotations which have no fixed points implying $L(T)=0$. There are 2 reflections which fix two vertices of index 1. There are 2 reflections which fix two edges of index 1. All 4 reflections have Lefschetz number $L(T)=2$.