P212121

A point group is a set of symmetry elements that are related to each other about a fixed point. A symmetry element is a point, line or plane around which the symmetry operation is performed. A symmetry operation is the process of transforming into a state indistinguishable from the starting state.

The following link provides a list of the 32 available point groups in Hermann–Mauguin notation. The 32 point groups are generated by performing the symmetry operations listed below:

Note: I am writing the dash over each number as ‘bar’ (how it is pronounced).

Point groups can contain the following operations:

1) Rotation:

• Symbol: 1,2,3,4,6
• Explanation: The number of times the item repeats in 360 degrees
• Example: A three fold requires three rotations to return to its starting position (360 degrees /3; n=3 for a threefold axis)

2) Reflection

• Symbol: m or 2bar
• Explanation: Created by a reflection across a plane
• Example: Place a hand above and below a table with the fingers pointing in the same direction. The table represents the mirror plane while your hands are a mirror image of each other.

3) Inversion

• Symbol: 1bar
• Explanation: An item is projected through a point that defines a center of symmetry
• Example: Put your hands in a separated prayer position, then flip one hand so that your fingers point downward

4) Rotoinversion

• Symbol: 3bar, 4bar, 6bar
• Explanation: rotate followed by an inversion
• Example: Step 1) Think of a fourfold rotation, move to the first position (90 degrees away); Step 2) inversion about the origin

A reflection or an inversion changes the handedness of the molecule and is, therefore, not very common in macromolecular crystallography (most amino acids are L not D in configuration).

I find visuals of point groups to be quite helpful. This a collection of all the point groups as movies. If the movies are rotating a bit fast for you, here are pictures of the point groups.

There is a mineralogy website that does a good job of covering symmetry operations. I also came across this pdf that covers point groups and introduces symbols of various operations.