For the following permutation in $S_9$, compute the sign, order and give cycle decomposition into disjoint cycles

Question

For the following permutation in $S_9$, compute the sign, order and give cycle decomposition into disjoint cycles, justifying your steps.

$\sigma \in S_9$ with

$$\sigma(1)=6, \sigma(2)=4, \sigma(3)=2, \sigma(4)=5, \sigma(5)=3, \sigma(6)=8, \sigma(7)=7, \sigma(8)=9, \sigma(9)=1 $$

My current solution

$(1, 6, 8, 9)(2, 4, 5, 3)$

Therefore $sgn(\sigma)=+1$

Not sure how do find the order, thanks in advance my knowledge on this topic is poor.

Answer 1

In general, the order of a permutation equals to the $lcm$ of the lengths of its disjoint cycles. (the cycles must have no common elements of course)