C4 carries 4 symmetry operations in 4 classes, around a principal axis of order 4. A character table of the group therefore has 4 columns and as many irreducible representations.
It contains only proper rotations, so a molecule in it is chiral and optically active. It leaves a direction invariant, so a molecule in it may carry a permanent dipole.
In Hermann-Mauguin notation it is 4, and it is one of the crystal classes of the tetragonal system. A diffraction pattern from a crystal in it shows the Laue symmetry 4/m.
Molecular point groups with their operations and character tables, the 230 space groups with a cell builder, and the 17 wallpaper and 7 frieze groups drawn live. The tool needs JavaScript, and WebGL2 for the 3D views; these are the pages it offers: