Analysis:
To determine O[Z(G)] and the number of conjugate classes of G, we need to understand the properties of the center of a group and conjugacy classes.

Center of a Group:
The center of a group G, denoted by Z(G), is the set of elements that commute with every element in G. In other words, Z(G) = {x ∈ G | xg = gx for all g ∈ G}.

Conjugacy Classes:
The conjugacy class of an element a in a group G, denoted by [a], is the set of all elements in G that are conjugate to a. In other words, [a] = {gag⁻¹ | g ∈ G}.

Solution:
1. Order of the Center (O[Z(G)]):
- The order of the center of a group G, denoted by O[Z(G)], is the number of elements in Z(G).
- For a non-abelian group G of order pq, where p is a prime number, we need to determine O[Z(G)].

2. Number of Conjugate Classes:
- The number of conjugate classes of G is denoted by |G|.
- We need to determine the number of conjugate classes of G.

Step 1: Order of the Center (O[Z(G)]):
- Since G is a non-abelian group, Z(G) is non-trivial (not just the identity element).
- By the class equation theorem, we know that the order of the center of a group divides the order of the group. Therefore, O[Z(G)] must divide pq.
- Since Z(G) is non-trivial, it cannot have only the identity element.
- Also, Z(G) cannot have order pq because it would make G abelian.
- Therefore, O[Z(G)] can be either p, q, or pq.

Step 2: Number of Conjugate Classes:
- The number of conjugate classes of G, denoted by |G|, is equal to the index of the center in G.
- By the class equation theorem, we know that the sum of the sizes of the conjugate classes is equal to the order of the group.
- Since G is non-abelian, Z(G) is a proper subgroup of G.
- Therefore, |G| is greater than 1.
- By the Orbit-Stabilizer theorem, the index of the center in G can be expressed as |G| = |G : Z(G)| = |G|/|Z(G)|.
- Since |G| is greater than 1 and |G| divides pq, |G| must be either p, q, or pq.

Summary:
- For a non-abelian group G of order pq, where p is a prime number:
- The order of the center (O[Z(G)]) can be p, q, or pq.
- The number of conjugate classes (|G|) can be p, q, or pq.