How would you combine the permutation
$\displaystyle (3,4)(2,4)(1,2)$
?
Trace what happens to each number:
The first two cycles have no "1" so 1 changes to 2:1->2.
In the second cycle, 2 changes to 4: 2->4.
In the first cycle, 3 changes to 4 and then in the second cycle 4 changes to 2, which, in the third cycle, changes to 1: 3->1.
In the first cycle, 4 changes to 3 which does not then change: 4->3.
In the basic notation that would be $\displaystyle \left(\begin{array}{cccc}1 & 2 & 3 & 4 \\ 2 & 4 & 1 & 3\end{array}\right)$.
an alternate, but equivalent formulation (and here i am following the opposite convention as HallsofIvy, multiplying right-to-left (like with functional composition)): perform (1 2), which is the function:
1-->2
2-->1
3-->3
4-->4.
next, perform the transposition (2 4):
1-->2-->4
2-->1-->1
3-->3-->3
4-->4-->2.
finally, we apply (3 4):
1-->2-->4-->3
2-->1-->1-->1
3-->3-->3-->4
4-->4-->2-->2, which is the permutation (1 3 4 2).
if, as Halls did, you multiply left-to-right (do (3 4) first, then (2 4), then (1 2), you wind up instead with (1 2 4 3)).
unfortunately for all of us, permutation multiplication is defined different ways, in different texts.