Ration by 0 degrees, i.e. the identity map.

Ration by 90 degrees clockwise.

Ration by 2*90 = 180 degrees clockwise.

Ration by 3*90 = 270 degrees clockwise.

Reflection through a horizontal axis (also called complement).

Reflection through a vertical axis (also called reverse).

Reflection through the main diagonal (also called inverse).

Reflection through the anti-diagonal.

The dihedral group of order 8 (the symmetries of a square); that is,

d8 = [r0, r1, r2, r3, s0, s1, s2, s3]

The Klein four-group (the symmetries of a non-equilateral rectangle); that is,

klein4 = [r0, r2, s0, s1]

orbit fs x is the orbit of x under the group of function fs. E.g.,

orbit klein4 "2314" == ["1423","2314","3241","4132"]

symmetryClasses fs xs is the list of equivalence classes under the action of the group of functions fs.

Symmetry classes with respect to D8.

Symmetry classes with respect to Klein4

rotate = r1 = inverse . reverse

The complement of the given permutation: if v = complement u then v `at` i = n - 1 - u `at` i.

The reverse of the given permutation: if v = reverse u then v `at` i = u `at` (n-1-i).

The group theoretical inverse: if v = inverse u then v `at` (u `at` i) = i.