Describe the subgroup $K\leq S_4$ of order 8 How do I construct the subgroup $K$ (a subgroup of $S_4$ of order $8$) ?

There are 3 such subgroups. Actually they are Sylow $2$-subgroups of $S_4$ and all of them are isomorphic (moreover, conjugate in $S_4$). For example, take $K=\langle (12), (13)(24) \rangle$ (the group generated by permutations $(12)$ and $(13)(24)$). 
More formally, $K \cong C_2 \wr C_2$ is the wreath square of $C_2$ (cyclic group of order $2$).
