Molecular Symmetry and Group Theory 
The C_{3v} character table is given opposite: 
 
Note that the numbers in a character table are not necessarily restricted to 1 or 1. We shall consider the meaning of the character 2, which describes the effect of the identity operation for the irreducible representation labelled E, in Workshop 4. 
Suppose we have the following reducible representation (reducible representations are normally given the label Γ):
E  2C_{3}  3σ_{v}  

Γ  4  1  2 
This may be reduced to its constituent irreducible representations by applying the reduction formula:
n(A_{1})  —  is the number of times the irreducible representation A_{1} occurs in the reducible representation, 

h  —  is the order of the group i.e. the total number of operations in the group, 
Σ  —  the sum is taken over all classes, 
χ_{R}  —  is the character of the reducible representation, 
χ_{I}  —  is the character of the irreducible representation, 
N  —  number of symmetry operations in the class. 
Fortunately this formula is much easier to use than to remember! Consider the reducible representation given above. For A_{1} we apply the following procedure:
E  2C_{3}  3σ_{v}  

Γ  4  1  2 
A_{1}  1  1  1 
1×4×1  1×1×2  1×(2)×3 
4  2  6 
Full Example of A_{1}:

 
Repeat the procedure above for A_{2} and E, and thus determine the composition of Γ.
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