A basic question regarding semidirect product of finite groups.

Question

I have a basic question regarding the semidirect product $C_m\rtimes C_k$ of two finite cyclic groups.

Does the semidirect product $C_m\rtimes C_k$ represents a specific group or rather a family of groups satisfying special relations? To sharpen my question, is it true that when we say

"Let be $G$ the group $C_m\rtimes C_k$..."

we refer to a specific group ? My confusion stems from the fact that by

https://en.wikipedia.org/wiki/Semidirect_product

the $C_m\rtimes C_k$ is define by the presentation $$ C_m\rtimes C_k=\langle a, b\mid a^m=1, b^k=1, b^{-1}ab=a^e\rangle, $$ which, to my understanding, defines a specific group.

Sorry for such basic question but I'm totally confused.

Answer 1

In general, for any two groups $N$ and $H$, the family of groups $N\rtimes H$ (not to confuse with isomorphism classes) is in bijection with the set ${\rm Hom}(H,{\rm Aut}(N))$. The description of $C_m\rtimes C_n$ in the wikipedia article is utilising the fact ${\rm Aut}(C_n)$ is in bijection with the set of numbers in ${\mathbb Z}/n{\mathbb Z}$, co-prime to $n$.