Group of isometries of the CC-plane (Q1047516)

Let \(\mathbb{R}^2_c=(\mathbb{R}^2,d_c)\) be the Chinese Checkers plane, where \(d_c(X,Y)= \text{max}\{|x_1-x_2|,|y_1-y_2|\}+({\sqrt 2}-1)\text{min}\{|x_1-x_2|,|y_1-y_2|\}\) for \(X=(x_1,y_1),\, Y=(x_2,y_2)\in \mathbb{R}^2\). The authors show that the group of isometries of \(\mathbb{R}^2_c\) is the semi-direct product of the dihedral group \(D_8\) and \(T(2)\), where \(D_8\) is the (Euclidean) symmetry group of the regular octagon and \(T(2)\) is the group of all translations of the plane.