The sliders for operations 1, 2, and 3 perform actions of the symmetry group

, as indicated by labels above the three arrows. The group

is isomorphic with the symmetric group

and the dihedral group

. You can explore the results of applying successive operations 1 and 2 of the symmetry group

, using tricolored equilateral triangles. The same result can be obtained in a single step using some operation 3. For example, the operation

followed by

gives the same result as the operation

. In standard group theory notation, this is written right-to-left as the symbolic product

. Note that

, showing that symmetry operations do not, in general, commute.