Direct product of two semi-direct products

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The discussion focuses on extending the argument for the group G, defined as the semi-direct product of two cyclic groups, specifically when the primes p and q are congruent to 1 modulo 3. Participants clarify the definitions of C_p and C_7:C_3, emphasizing that C_p represents a cyclic group of order p, where p is a prime. The semi-direct product C_7:C_3 is highlighted as a specific example within this context. The main inquiry seeks assistance in applying the established findings to the case where both primes are congruent to 1 modulo 3. The conversation underscores the importance of understanding the properties of these groups in relation to their structure.
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Homework Statement
I need to find the number of elements and conjugacy classes for the direct product.
Relevant Equations
$$G=(C_7:C_3\ )\times(C_{13}:C_3\ )$$
After finding the number of elements for this group, how do I extend the argument to $$p,q\equiv1\left(mod\ 3\right)$$, where $$G=(C_p:C_3\ )\times(C_q:C_3\ )$$Any help appreciated.
 
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What is ##C_p## and what is ##C_7:C_3##?
 
$$C_7 : C_3$$ is a semi-direct product.

$$C_p$$ is as described. A prime, congruent to 1 mod 3.
 
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