This is more of an afterthought to my earlier question here.
The interaction among the vertices (shown in disjoint cycles of $S_n$) are derived from the underlying operations (like: reflection, rotation) to yield group elements being more than vertices in cycles.
Say, $S_3$ has three vertices in cycles, but has $6$ as order.
But, the number of such actions (as in $S_3$ too) is shown by different set of cycles. Say $(1)$ for identity, $(123)$ for $R_1$, and so on.
As answer to question is $a.b= 1000$ , then only $a+b$ vertices are there, and generate $1000$ maximum group actions (or their combinations). These $1000$ are the group elements. While vertices are just a set on which actions apply.
If so, what is the exact relationship between the set of vertices and group elements?
In $S_{133}$ with $a=8, b=125$, how $a+b= 133$ vertices can possibly generate $1000$ group elements is unclear (have only two operations known over symmetric group: reflection, rotation).
The number of elements of $S_n$ is
$$n!=n\times (n-1)\times\dots\times 2\times 1.$$
Remember that $n$ is the number of elements of the set $S_n$ acts on by definition, which is typically $\{1, \dots, n\}$.