When we consider the group $\mathbb{Z}_5 \times \mathbb{Z}_5$, a pair like $\{(0,1),(1,0)\}$ can generate the group. Also $\{(0,2),(3,0)\}$ can generate the group.
When we consider $(\mathbb{Z}_5 \times \mathbb{Z}_5) \rtimes \mathbb{Z}_3$, a set $\{((0,1),0), ((0,0),1)\}$ as well as $\{((0,2),0),((0,0),1)\}$ can be generating sets. Likewise there are many choices for generating elements. When we look at the first element in both of the above sets, we can observe their $y$- coordinates are 0, so that they must be of order 5. Hence, even if one doesn't specify that the above sets are generating sets we can identify that first two elements are possible candidates for order 5 generating elements. Similarly, we can identify the second two elements are order 3 generating elements.
As an example, if we have a sequence of elements,
$\{((0,3),0), ((0,0),1), ((2,1),0), ((1,1),2), ((2,3),2), ((0,0),2)\}$
By observing the $x$ and $y$ coordinates we can say that $((0,0),1), ((0,0),2)$ are possible candidates for order 3 generating elements and $((0,3),0), ((2,1),0)$ are possible candidates for order 5 generating elements (if I'm to arbitrarily select elements for a generating set with order 3 and order 5 generating elements).
Can we determine the generating or non-generating nature of other elements in the above sequence similarly by observation?
And if we have any semidirect product $H \rtimes_{\phi} K$ where $H,K$ are finite groups, the elements of the group are from $H \times K$, when written as ordered pairs like above.
If we are given a sequence of arbitrary elements like above can we always determine the possible candidates for generating elements by observing $x$ and $y$ coordinates like above (Suppose we know the groups $H,K$ and their orders)?