C6h carries 12 symmetry operations in 12 classes, around a principal axis of order 6. A character table of the group therefore has 12 columns and as many irreducible representations.
It contains an improper operation, so no molecule in it is chiral. No direction is left invariant, so a molecule in it has no permanent dipole. It contains the inversion, so its vibrations and its electronic states divide into g and u, and no vibration is both infrared and Raman active.
In Hermann-Mauguin notation it is 6/m, and it is one of the crystal classes of the hexagonal system. A diffraction pattern from a crystal in it shows the Laue symmetry 6/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: