What meaning can be given to a permutation to the power of another permutation? Say you have 2 permutations, f and g, how would one calculate $g^f$? Also can you multiply these permutations and how?

Usually in group theory $g^f$ is used notation for conjugate of $g$ by $f$, i.e. $g^f := f^{-1}gf$. You can calculate this product in the following manner. If $g = (a_1a_2\cdots a_n)$, then $g^f = f^{-1}gf = (f^{-1}(a_1)f^{-1}(a_2)\cdots f^{-1}(a_n)$).
On the otherside multiplication of permutations is composition of them. To find the actual product, you would need to do term by term caclulations, i.e. $g \cdot f (a) = g(f(a))$
