I have a given $\sigma$ which is:
$$\left({1\atop5}{2\atop8}{3\atop3}{4\atop4}{5\atop6}{6\atop1}{7\atop7}{8\atop2}\right)\in S_8$$
I need to find the order of this permutation, as it is necessary to solve $\sigma^{122}$
My feeling is that once you have the order, the solution is to find the remainder of the power following the division by the order of the permutation.
e.g
If the order is $5, \frac{122}{5} = 24.5$
$24\cdot 5 = 120$ meaning that $σ^{122} = σ^{120} \cdot σ^2 = σ^2$
Indeed, as mentioned in a comment, the decomposition as a product of disjoint cycles is $$\sigma=(1\,5\,6)(2\,8).$$ As these cycles commute, the order of $\sigma$ is the l.c.m. of the orders of the cycles, which is respectively $3$ and $2$. Now, $$\sigma^{122}=(1\,5\,6)^{122} (2\,8)^{122}=(1\,5\,6)^{122\bmod 3} (2\,8)^{122\bmod 2}=(1\,5\,6)^{-1}=(1\,6\,5).$$