Rotations preserving an axis ($SO(2)⋊C_2\cong O(2)$)

The subgroup of three-dimensional proper rotations preserving a chosen unoriented axis consists of all rotations about it and all half-turns about perpendicular axes. It is isomorphic to $O(2)$; the continuous molecular point-group notation is $D_{\infty }$. This uncountable group is different from the abstract countable infinite dihedral group $\mathbb{Z}⋊C_2$.