Symmetric groups having an element of order $\geq 8$

Question

Is it true that all symmetric groups $S_5,S_6,\cdots,S_{10}$ have an element of order $d$ with $d\geq 8$ or $d=6$?

Answer 1

In $S_5$ we have $(1,2)(3,4,5)$. In the other groups, we have $(1,2,3,4,5,6)$.

Answer 2

Since the order of a product of disjoint cycles is the least common multiple of the lengths of the cycles and $n \geq 5$, we know that $(12)(345) \in S_{n}$ and $|(12)(345)| = \text{lcm}(2,3) =6$. Note there wasn't anything special about $(12)(345)$, it just happened to be the product of 2-cycle and 3-cycle disjoint from one another.

Also recall that every permutation can be written as a product of disjoint cycles. So if $n = 5,6,7$, is it possible to have an element of order 8?