Editing Hexomino (section)

==Symmetry==
The figure above shows all 35 possible free hexominoes, coloured according to their [[symmetry group]]s:
* The twenty grey hexominoes have no [[symmetry]].  Their symmetry group consists only of the [[identity function|identity mapping]].
* The six red hexominoes have an axis of [[Reflection symmetry|mirror symmetry]] parallel to the gridlines.  Their symmetry group has two elements, the identity and a reflection in a line parallel to the sides of the squares.
* The two green hexominoes have an axis of mirror symmetry at 45° to the gridlines.  Their symmetry group has two elements, the identity and a diagonal reflection.
* The five blue hexominoes have point symmetry, also known as [[rotational symmetry]] of order 2.  Their symmetry group has two elements, the identity and the 180° rotation.
* The two purple hexominoes have two axes of mirror symmetry, both parallel to the gridlines (thus one horizontal axis and one vertical axis).  Their symmetry group has four elements. It is the [[dihedral group]] of order 2, also known as the [[Klein four-group]].

If reflections of a hexomino are considered distinct, as they are with one-sided hexominoes, then the first and fourth categories above would each double in size, resulting in an extra 25 hexominoes for a total of 60. If rotations are also considered distinct, then the hexominoes from the first category count eightfold, the ones from the next three categories count fourfold, and the ones from the last category count twice. This results in 20 × 8 + (6 + 2 + 5) × 4 + 2 × 2 = 216 fixed hexominoes.