Let $G=\langle(12),(123),(4567)\rangle≤ S_7$. How would I show that $G\cong S_3 × C_4$?

Question

Let $G=\langle(12),(123),(4567)\rangle≤ S_7$. How would I show that $G\cong S_3 × C_4$?

We've been working a lot on isomorphisms recently, and i definitely understand how to prove them (f(a)f(b)=f(ab), one-to-one, onto). However, I'm having some trouble with these two groups. I was thinking I would start with the properties of either group? or would it be better to just define a function?

Answer 1

HINT: Consider what generates $S_3$, in particular, since you have a 2 cycle and a 3 cycle with the same numbers, those generate an isomorphic image of $S_3$. Consider what generates $C_4$, which is a single element of order 4, which you have in a 4 cycle.

Define your map sending generators to generators, and then show that the relationships hold (Which they will, because the thing generating your 4 cycle commutes with your things generating $S_3$, since the numbers dont overlap)