When we consider the finite group $\mathbb{Z}_p \times \mathbb{Z}_p$, where $p$ is a prime, $p>2$, the set with the pair of elements $\{(0,1), (1,0)\}$ can generate the group. Moreover, a set $\{(1,0),(1,2)\}$ can also generate the group. Hence, there are many pairs of elements which can act as generating elements.
Consider the group $(\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes_{\phi} \mathbb{Z}_q$. A set $\{t,s\}$, where $|t|=p$ and $|s|=q$ is a minimal generating element set for some $p,q$ and $\phi$ values (Let $p$ and $q$ be distinct primes, $p,q>2$).
Under a certain $\phi, p, q$, the $t,s$ elements were,
$t=((1,0),0),\,\, s=((0,0),1)$
These generating elements can be used to draw a Cayley graph of the group.
However, if I take $t=((3,0),0)$ and draw the graph (under the same $\phi$. Suppose $\phi$ is defined using a general formula like, $\phi(t^i)=t^{i+1}$. (This is only an example). Then I can apply $\phi(t)$ to any $t$ value), the graph is the same as earlier graph if I forget the vertex lables.
Then can I bring an argument here, that I can use any element of order $p$ as $t$ and any element of order $q$ as $s$ in the generating set?
Thanks a lot in advance.
Summarized idea of the answer of above question:
For a group $G = (\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes_{\phi} \mathbb{Z}_q$,
$\phi: \mathbb{Z}_q \rightarrow Aut(\mathbb{Z}_p \times \mathbb{Z}_p) $
defines the semidirect product. Let $S=\{s,t\}$, where $|s|=q$ and $|t|=p$ denote an irredundant generating set of $G$ and let $\{t,u\}$, where $|u|=p$ be a basis of $\mathbb{Z}_p \times \mathbb{Z}_p$. ($\phi_s(t) = s^{-1}ts =u$).
For an element $\phi(s)=\phi_s \in Aut(\mathbb{Z}_p \times \mathbb{Z}_p)$, in defining the automorphism,
$\phi_s(t^{i} u^{j})$ using the generating elements $t,u \in \mathbb{Z}_p \times \mathbb{Z}_p$, we can choose different candidates of order $p$ as $t$ and $u$, chosen appropriately.
i.e. as an example,
$t=((0,1),0), u=((1,0),0)$ is one choice, but also $t=((0,3),0), u=((1,0),0)$ can be another choice. Likewise, there can be several choices for the generating elements.