Note that satisfy the relations and and so you can easily verify that the map (where is the Quaternion group) defined by and is a homomorphism, and since it maps generators to generators we see that it surjects onto , but clearly and so . Now, since can be covered by we know that but since you must have that . But, since (as can easily be checked) are distinct elements of we may conclude that and moreover that is an isomorphism (since a finite surjection between two sets of equal cardinality must be a bijection).