Rotations preserving an axis
= Rotations preserving an axis
{title2=$SO(2)\rtimes C_2\cong O(2)$}
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 $\mathbb Z\rtimes C_2$.