Finding $\gamma \in S_7$ that satisfies $\gamma^4=(3412675)$

Question

I have proved it using an arbitrary $\tau=(1234567)$, taking it to the fourth power, and finding that $\tau^4=(1526374)$.

Can I just see the pattern of where each element went and match it to the $\gamma^4$?

Answer 1

Hint: Since $\gamma^4$ is a $7$-cycle, so is $\gamma$. Then, what is $(\gamma^4)^2$?

Answer 2

Travi's solutions is pretty neat. Alternatively observe a cycle can only break into smaller cycles when exponentiationg, so $\gamma$ is a $7$ cycle.

What then build the $7$ cycle:

$(1***5**)\rightarrow (12**5**)\rightarrow (12**56*)\rightarrow (123*56*)\rightarrow (123*567)\rightarrow (1234567)$