We will first choose the oxygen atom at the center of the unit cube as the origin and then label the six atoms as:

Atom [1]= Oxygen atom at (0,0,0);
Atom [2]= Oxygen atom at (a/2)(1,1,1) where a is the size of the unit cube;
Atom [3]= copper atom at (a/4)(1,1,1);
Atom [4]= copper atom at (a/4)(-1,-1,1);
Atom [5]= copper atom at (a/4)(1,-1,-1); and
Atom [6]= copper atom at (a/4)(-1,1,-1).

(a) To determine the characters for the group G we have to apply first the symmetry elements of the T_d group to the six atoms and in each count the number of atoms which are are unchanged by the operation. For example, the symmetry operation C₂(z) (a 2-fold rotation about the z-axis) will change [2] to the position (a/2)(-1,-1,1) which differs from its original position by a lattice vector. Because of the periodicity of the lattice we consider [2] as unchanged. On the other hand atoms [3] and [4] obviously will interchange their positions and similarly for [5] and [6]. Thus the total number of atoms unchanged by C₂(z) is 2. Thus the character of C₂(z) is 2.

Similarly the 3-fold rotation (C₃) about the [111] axis will leave the oxygen atoms unchanged while permuting the position of all the copper atoms except [3]. Thus its character is 3

The rotation m_d into the [110] plane will leave atoms [1], [2], [3] and [4] unchanged while interchanging atoms [5] and [6] making its character equal to 4.

The readers should show that the character of the class {S₄}is 2.

Next we consider the symmetry operation I involving inversion followed by the translation of (a/2)(1,1,1). Under this operation the two oxygen atoms interchange their positions while [3] is unchanged. . Under I the position of atom [4] is changed to (a/4)(3,3,1). Applying the lattice translation (a/4)(-4,-4,0) [4] is returned to its original position. Similarly the other copper atoms are unchanged by I so its character is 4.

In summary, the characters of G for the operations of the T_d group and I are given by:

{E}	{C2}	{S4}	{md}	{C3}	{I}
6	2	2	4	3	4

(b) Based on these characters and Table 2.16 we can already determine uniquely the that G can be reduced to the following representations: 2G₁⁺ +G₂^-+G₅⁺. [Note the errors in the book which have now been corrected in the errata]

(c) Again using Table 2.16 we can show that: (2G₁⁺ +G₂^-+G₅⁺)xG₄^-=G₂^-+G₃^-+3G₄^-+G₅^-+G₅⁺

Reference: K . Huang: The long wave modes of the Cu₂O lattice. Festschrift für Physik, 171, 213-225 (1963)