Is $S_{10}$ generated by the group of $6$ cycles?

Question

Is $S_{10}$ generated by the subgroup of $6$ cycles?

I'll denote that subgroup by $H$.

We know that by conjugation permutations remains with the same cycle structure, and therefore $H$ is closed to conjugation. It is easy to see from here that $H$ is normal subgroup.

The only normal subgroup of $S_{10}$ is $A_{10}$, therefore $H$ must be either $S_{10}$ or $A_{10}$.

How can i determine which one from here?

Are the specific numbers $10,6$ even relevant?

Answer 1

Each $6$-cycle has sign equal to $-1$. Therefore, $H\not\subset A_{10}$.