I'm trying to classify the groups of a certain order and have shown that a group $G$ with that order can be expressed as the semidirect product of a normal subgroup $N$ $\cong$ $C_n$ $\times$ $C_m$ and a cyclic subgroup $H$ $\cong$ $C_p$. $n$, $m$ and $p$ are all different primes.
$φ:H→Aut(N)$ $\cong$ $Aut($$C_n$ $\times$ $C_m$$)$ $\cong$ $C_{n-1}$ $\times$ $C_{m-1}$ , by Euler's totient function.
That means I'm looking for homomorphisms $φ:$$C_p$ → $C_{n-1}$ $\times$ $C_{m-1}$. Using some properties about element orders, I defined each possibility by fixing a generator of $C_p$, so I know what each homomorphism does.
Now I have to make group presentations for each homomorphism, but I don't know how to intepret the result I got from another answer that
$G\rtimes_\phi H \;=\; \langle X, Y \mid R,\,S,\,yxy^{-1}=\phi(y)(x)\text{ for all }x\in X\text{ and }y\in Y\rangle \tag{1}$
since in this case each homomorphism gives an output belonging to $C_n$ $\times$ $C_m$ and is an element with two coordinates. I understand that $R$ and $S$ are the relations of the cyclic groups.
Any help would be greatly appreciated!