C3v group operations
S. M. Blinder · Acervo Digital da Universidade Estadual Paulista (Universidade Estadual Paulista) · 2009
The sliders for operations 1, 2, and 3 perform actions of the symmetry group C_3v, as indicated by labels above the three arrows. The group C_3v is isomorphic with the symmetric group S_3 and the dihedral group D_3. You can explore the results of applying successive operations 1 and 2 of the symmetry group C_3v, using tricolored equilateral triangles. The same result can be obtained in a single step using some operation 3. For example, the operation C_3 followed by σ_1 gives the same result as the operation σ_2. In standard group theory notation, this is written right-to-left as the symbolic product σ_1C_3=σ_2. Note that C_3σ_1=σ_3, showing that symmetry operations do not, in general, commute