Let \(f\) and \(g\) be the following permutations in \(S_{7}\) :

\[ f=\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 1 & 5 & 7 & 2 & 6 & 4 \end{array}\right), \quad g=\left(\begin{array}{ccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 1 & 7 & 6 & 4 & 5 & 2 \end{array}\right) . \]

Write down in cycle notation the permutations \(f, g, g^{2}, g^{3}, f \circ g,(f \circ\) \(g)^{-1}\) and \(g^{-1} \circ f^{-1}\).
What is the order of \(f\) ? What is the order of \(f \circ g\) ?