What can the multiplication table tell us about the representation?

[sadness]

In the appendix B of Goldstein's classical mechanics (3rd edition), the authors discussed the dihedral group and said:

"Notice how the group elements in class 3 involve only \sigma_1 and \sigma_3. Thus, they are independent of the matrices I and \sigma_2, as is expected from the structure of the multiplication table. However, since each representation has an identity element, there is no simple association between classes and representations."

Why does the structure of the multiplication table indicate this independence? And what does the last sentence mean? I've attached the multiplication table and the representations.

 

[fresh_42]

This is just a clumsy way to say that the Dihedral groups are semidirect products $\mathbb{Z}_2 \ltimes \mathbb{Z}_n\,.$

Elements of order $2$ do not occur in (the multiplication table of) $\mathbb{Z}_3 \triangleleft D_3$, and there is no "simple association between classes and representations" means, that the product isn't direct: $\mathbb{Z}_2$ operates non trivially on $\mathbb{Z}_3$.