The package contains the command Symmetry. By typing Symmetry[] without an argument, a list of all point group symbols known to the package is printed:

23, T, 2/m~3, m~3, m3, ~T, Th, 432, K, O, ~43m, KT, Td, 4/m~32/m, m~3m, m3m, ~K, Oh, I, Y, ~I, ~Y, C(2n)C(n), D(n)C(n), D(2n)D(n), C(n_Odd)i, C(n)h, D(n)d, D(n)h, C(n)v, C(n), ~C(n), D(n), ~D(n), S(n_Even), (n_Even)/m2/m2/m, (n_Even)/mmm, (n_Even)/m, (n_Even)22, (n_Odd)2, ~(n_Even)2m, ~(n_Odd)2/m, (n_Even)mm, (n_Odd)m, ~(n), (n)

Here, (n) may be any integer number, (n_Odd) any odd integer number, and (n_Even) any even integer number. The elements of this list are strings, so they should be given in quotes when used as arguments in commands. A tilde before a symbol is equivalent to a bar over the corresponding Schönflies symbol; for example, the Schönflies symbol is written as "~3". An overview of all point group symmetries and their symbols can be found in (Rabson D.A., Mermin N.D., Rokshar D.S., Wright D.C., "The space groups of axial crystals and quasicrystals", Rev. Mod. Phys. 63(3), (1991), 699-733.). For a list of the symbols of the crystallographic point groups, see here. Please note that the package knows not only the crystallographic point groups (compatible with translational symmetries), but all point groups, for example the icosahedral point groups I and Y.

The command Symmetry[g] gives a list of all symmetry transformations of the point group g, for example when typing Symmetry["~3"] the following list is returned:

{{{0, 0, 1}, 0, 1}, {{0, 0, 1}, 2 Pi/3, 1}, {{0, 0, 1}, 4 Pi/3, 1}, {{0, 0, 1}, 0, -1}, {{0, 0, 1}, 2 Pi/3, -1}, {{0, 0, 1}, 4 Pi/3, -1}}

The result is a list of all symmetry transformations of the point group . Here, a transformation is given in the form {{x, y, z}, f, i}, where the vector {x, y, z} defines the axis of rotation, f is the angle of rotation, and i = +1 indicates a proper rotation, and i = -1 a rotoinversion (a rotation with a subsequent inversion). The vector components refer to a rectangular Cartesian coordinate system. Thus, the transformation {{0, 0, 1}, 0, 1} is the identity operation, and {{0, 0, 1}, 0, -1} is a pure inversion. In this notation, a reflection in a plane perpendicular to the vector {x, y, z} is defined by {{x, y, z}, p, -1}.

The command ShowSymmetry> visualizes all the symmetries of a given point symme-try group, for example ShowSymmetry["~T"] (tetrahedral group) displays the following image:

Finally, for even better visualization, one can rotate the wjole image and produce a movie. For example, the command ShowSymmetry["D6h", Rotate->{{0,1,0},{0, 19 p/20}}] generates the following movie displaying the symmetries of the point group D_6h