I'm sure there is a quick and straightforward method for answering this question. But I can't find it in my textbook.

Here's what I know:

$\displaystyle \sigma \tau$ in the array notation would be:

$\displaystyle \begin{bmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 &8&9&10&11\\6 & 4 & 11 & 9 & 1 & 7 & 3 & 5 & 8 &2&10\end{bmatrix}$

and can be written as a product of disjoint cycles:

$\displaystyle (1,6,7,3,11,10,2,4,9,8,5)$

Since it's only one cycle of length 11, the order is lcm(11)=11

I'd be thankful if anyone could show me what method to use to find $\displaystyle (\sigma \tau)^{9000}$ and $\displaystyle (\sigma \tau)^{-21}$.