Find the smallest $n$ such that $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ is isomorphic to a subgroup of $S_n$

The smallest $n$ is $6$:
1. $A$ is isomorphic to $\langle(1,2),(3,4),(5,6)\rangle$.
2. For $n=4,5$ the only subgroup of order $8$ which $S_n$ does contain is the dihedral group $D_4$ (and its conjugates, being a $2$-Sylow subgroup).