S6 carries 6 symmetry operations in 6 classes, around a principal axis of order 6. A character table of the group therefore has 6 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 -3, and it is one of the crystal classes of the trigonal system. A diffraction pattern from a crystal in it shows the Laue symmetry -3.
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: