This group is called the generalized dihedral group of Z, Dih(Z), and also D ∞. It is a semidirect product of Z and C 2. It has a normal subgroup of index 2 isomorphic to Z : the translations. Also it contains an element f of order 2 such that, for all n in Z, n f = f n : the reflection with respect to the reference point, (0,1).