C3v Group Operations
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.
Snapshot 1: the group operations designated
Snapshot 2: , showing noncommutativity of group operations
Snapshot 3: every element has an inverse