This question is motivated by the answer to this one There is also another followup. The question there was " Given integers $m,n,k \gt 1$ construct permutations $x,y$ with $o(x)=m,o(y)=n$ and $o(x,y)=k.$" Derek Holt relates a very elegant solution using $PSL(2,q)$ for suitable odd $q$. It is nice in that it provides a solution quickly albeit not a small or obvious one. He illustrates with $m,n,k=10,12,15$ and comes up with a solution. I tried briefly to understand what was "really" going on from staring at the permutations without refernce to the method which gave them. I did not get very far. I will report that rearranging the underlying set the solution is

$x=$[[2, 82, 17, 98, 22, 34, 83, 13, 86, 56], [3, 50, 10, 42, 20, 19, 97, 8, 93, 41], [4, 43, 113, 75, 33, 95, 80, 63, 28, 100], [5, 21, 39, 118, 119, 9, 45, 117, 96, 30], [6, 32, 49, 35, 84, 120, 68, 110, 36, 60], [7, 62, 64, 74, 16, 40, 115, 67, 101, 109], [11, 26, 121, 44, 61, 94, 15, 48, 37, 87], [12, 59, 77, 73, 54, 69, 108, 107, 76, 52], [14, 47, 72, 105, 89, 122, 116, 51, 31, 81], [18, 65, 99, 27, 78, 111, 92, 102, 104, 55], [23, 24, 79, 106, 57, 112, 70, 25, 88, 53], [29, 90, 58, 66, 114, 103, 38, 71, 85, 91]]

$y=$[[1, 2, 57, 107, 109, 102, 93, 9, 120, 85, 72, 48], [3, 42, 11, 88, 26, 12, 53, 89, 91, 86, 14, 82],[4, 101, 68, 106, 80, 96, 118, 40, 17, 83, 35, 50], [5, 16, 75, 114, 67, 116, 122, 90, 30, 97, 20, 43], [6, 46, 47, 15, 95, 34, 23, 54, 74, 65, 19, 21], [7, 110, 69, 55, 105, 73, 78, 28, 64, 63, 81, 32], [8, 98, 18, 56, 87, 38, 104, 103, 115, 41, 94, 62], [10, 51, 117, 31, 52, 77, 60, 37, 49, 33, 61, 45], [13, 84, 36, 111, 79, 25, 71, 39, 22, 99, 66, 59], [27, 100, 29, 92, 112, 58, 76, 108, 70, 113, 44, 121]]

$xy=[[1, 2,\cdots, 15], [16, 17, \cdots, 30], \cdots, [106, 107, \cdots 120]]$

But that says little (to me). To be fair, it was not suggested that $10,12,15$ is a hard case, merely a good one for illustrating the method.

I will ask the specific question in a moment. The underlying questions are: What are the hard cases? When can there be solutions of special kinds? How big must $B$ be to have a solution in $S_B$? The meta question is: What is the right question?

I will say that in the solution above $x,y$ and $xy$ have **cycle structures** $C(x)=z_{10}^{12},C(y)=z_{12}^{10}$ and $C(xy)=z_{15}^8.$

So I propose the question:

For which triples of cycle structures $\alpha,\beta,\gamma$ are there permutations $x,y$ with $C(x),C(y),C(xy)=\alpha,\beta,\gamma?$

It is an open ended question. If a complete characterization is too much to hope for, then nice sufficient conditions would be ...nice. It is natural to feel that "often" there are solutions which can be written with the integers in the cycles have relatively long runs in order or reverse order. Perhaps with each cycle in $x$ intersecting just one or two of $y.$ That feeling might be mistaken, but when is it not? I will give a few solutions below which I would say have this (vaguely specified) nature.

The example above suggests to me that the perhaps the $PSL(q)$ method establishes that there are always solutions of structure $z_m^i,z_n^j,z_k^{\ell}$ for some $i,j,\ell$. I might be wrong. If not, then perhaps more can be said about $i,j,\ell.$ In this particular case of $m,n,k=10,12,15$ I also find this example with $i,j,\ell=3,1,2.$

$x=$[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34]]

$y=$ [[1, 11, 21, 22, 23, 24, 25, 34, 33, 32, 31, 16]]

$xy=$[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]]

I did not find and examples $z_{10}^i,z_{12}^j,z_{15}^{\ell}$ with smaller $i,j,\ell.$

I did find $x=$[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33]]

$y=$[[1, 11, 12, 13, 14, 15], [16, 17, 18, 19, 20, 21, 22, 23, 24, 33, 32, 31]] with the same $xy.$

Also $x=$[[1, 2, 3, 4, 5, 6, 7, 8, 9, 12]], $y=$[[1, 13, 14, 15], [10, 11, 12]] with

$xy=$[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]].

And Derek gave the minimal. $x=$[[1, 4, 6, 2, 3], [5, 7]],$y=$ [[1, 4, 2, 7], [6, 5, 8]] with $xy=$[[1,2,3,4,5],[6,7,8]].