Klein four-group

The Klein four-group has a representation as 2×2 real matrices with the operation being matrix multiplication:

On a Rubik's Cube the "4 dots" pattern can be made in three ways, depending on the pair of faces that are left blank; these three positions together with the "identity" or home position form an example of the Klein group.

In three dimensions there are three different symmetry groups that are algebraically the Klein four-group V:

The three elements of order two in the Klein four-group are interchangeable: the automorphism group of V is the group of permutations of these three elements.

The Klein four-group's permutations of its own elements can be thought of abstractly as its permutation representation on four points:

In the construction of finite rings, eight of the eleven rings with four elements have the Klein four-group as their additive substructure.