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For related videos, refer to the following links;

For the structure of the character table,    • Structure of the Character Table | Group T...  

For the physical significance of irreducible representation,    • Physical Significance of Irreducible Repre...  

For the significance of Mulliken symbols PART 1,    • Significance of Mulliken Symbols PART 1 | ...  

For the significance of Mulliken symbols PART 2,    • Significance of Mulliken Symbols Part 2 | ...  

For the mixed and closed functions in group theory,    • Mixed and Closed Functions in Group Theory...  

For the reducible representation Part 1,    • Reducible Representation | Physical Signif...  

In this lecture, we move from the physical meaning of reducible and irreducible representations to the essential mathematical tools of group theory.

We begin by understanding two fundamental concepts in molecular symmetry: order and class of a point group.

What does the order of a group really mean?
Why does the octahedral point group have 48 symmetry operations, while C2v has only 4?
What is the significance of representing order by h?

Then we dive into the deeper and more powerful idea of a class.

You’ll learn:

• What it means for symmetry operations to be equivalent
• The precise mathematical condition for two operations to belong to the same class
• How conjugation connects symmetry operations
• Why the 8 C3 axes in the octahedral point group form one class
• Why different sets of C2 axes in Oh belong to different classes
• What “structurally equivalent transformation” really means
Through clear examples using functions like (x, y, z) and orbitals like dxy, we explain how operations in the same class transform functions in the same structural pattern — even if the exact mapping is different.

We then connect this directly to character tables:

• Why all operations in the same class have the same character
• Why the number of columns equals the number of classes
• Why the number of irreducible representations equals the number of classes
• Why Oh has 10 classes
• Why C2v has 4 classes and no conjugacy

This lecture builds the conceptual foundation needed to truly understand character tables and symmetry analysis in spectroscopy, bonding, and molecular orbital theory.

If you're serious about mastering molecular symmetry and group theory in chemistry, this is a key step.