GATE 2024 CY – Question 19

Physical Chemistry · Group Theory: Symmetry elements and symmetry operations. Basic postulates of groups and group multiplication tables. Classes, reducible and irreducible representations of symmetry elements. Classification and labelling of molecular point groups. Symmetry based selection rules and their application to electronic and vibrational spectroscopy. Internal coordinates and symmetry labelling of vibrational modes; symmetry adapted linear combination of atomic orbitals (LCAO-MO); construction of hybrid orbitals using molecular point group symmetry. · 1 mark · Multiple choice

The order and the number of classes present in a group with the irreducible representations $A_1,A_2,B_1,B_2,E_1$, and $E_2$, are, respectively,

6 and 6
12 and 6
6 and 3
12 and 3

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Correct answer: (B) 12 and 6

Explanation

Use two facts about the irreducible representations (irreps) of a finite group.

**Order of the group.** The order $h$ equals the sum of the squares of the dimensions of the irreps:
$$h=\sum l_i^2 .$$
The one-dimensional irreps $A_1,A_2,B_1,B_2$ contribute $4\times1^2=4$, and the two-dimensional irreps $E_1,E_2$ contribute $2\times2^2=8$:
$$h=4+8=12 .$$

**Number of classes.** The number of conjugacy classes equals the number of irreducible representations, which is 6.

The order is 12 and the number of classes is 6 (option B; this is the group $C_{6v}$).