Permutation/1357642/Several properties/Exercise

We consider the permutation $\tau \in S_7$, given by the value table

$x$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$\tau (x)$	$1$	$3$	$5$	$7$	$6$	$4$	$2$

1. Determine the cycle representation of $\tau$, and the range of action.
2. Compute ${\tau }^3$ and the order of ${\tau }^3$.
3. Determine the inversions of $\tau$ and the sign of $\tau$.
4. Express $\tau$ as a product of transpositions, and determine again the sign of $\tau$

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