Find a cyclic group in size 6 in $S_5$

Question

I have the following question :

Find a cyclic group in size 6 in $S_5$.

Meaning $H=\langle f_1,f_1^2,f_1^3,f_1^4,f_1^5,f_1^6=e\rangle$

I wonder if there's a method to solving such questions.

I know that $f_1^6=e$, My question is more general assuming I need to find a cyclic group in size $1,2,3,..$ in $S_5$ should I just "test" until I find the right one I wonder if there's a smarter method to solve these kind of questions.

Any ideas?

Any help will be appreciated.

Answer 1

The cyclic subgroup generated by $(1 2)(3 4 5)$ has order $lcm(2,3)=6$.