I want to list all the cosets of $S_4$ in $S_5$, where $S_4$ means symmetric group on 4 elements. One way is try to do composition with all the elements of $S_5$ with $S_4$. A little better is eliminate the set of all elements of $S_4$ which are in $S_5$. Working further on this to me it appears that if I do composition with $4$ transpositions of $S_5$ with $S_5$ will be sufficient.
Question : How to listing the all cosets efficiently? ( without using GAP )
Take the set $M = \{1,2,3,4,5\}$. Then $S_4 \leq S_5$ is the stabilizer of $5$ under the standard action of $S_5$ on $M$. Thus, by the orbit stabilizer theorem, there is a bijection between $M$ and the cosets. Namely, the cosets are given by $$C_i = \{\pi \in S_5 \mid \pi(5) = i\}$$ for $1 \leq i \leq 5$.
There are different ways to write all elements of these sets down, depending on what you want them for.