Use the fact that the order of a permutation is the least common multiple of the lengths of its cycles.

The 7 elements must be partitioned into one of the following ways :

7

1+6

2+5

3+4

1+1+5

etc.

It's not hard to see that the least common multiple is maximized when using the partition 3+4=7, yielding a maximum order of . So the product of a three-cycle with a four-cycle has order 12 and that is maximum.