Q8 is not Isomorphic to any subgroup of S7
Problem The Quaternion group $G = Q_8$ is not isomorphic to any subgroup of $S_7$. Proof Let $A$ be a set with 7 elements. Any homomorphism $\alpha: G \to S_7$ can be viewed as a group action $G$ on $A$. For each $a \in A$, consider the size of its orbit $Ga$ and stabilizer $G_a$. Orbit-stabilizer theorem gives the following equality: $$ |Ga| = \frac{|G|}{|G_a|} $$ Since $Ga \subseteq A$, we have $|G_a| \leq 7$. Then ...