# C3v Group Operations

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.

C

3v

C

3v

S

3

D

3

C

3v

C

3

σ

1

σ

2

σ

1

C

3

σ

2

C

3

σ

1

σ

3