What is the order of the following subgroup $\langle (1 \ 2 \ \cdots \ n), (a \ b)\rangle$ of $S_n$?

Question

Let $n\geq 2$ be an integer, and consider $H=\langle (1 \ 2 \ \cdots \ n), (a \ b)\rangle\subset S_n.$

It is known that $H=S_n$ if and only if $b-a$ and $n$ are coprime.

Question. What is the order of $H$ in general ?

For the moment, I think I have managed to prove the following facts:

Let $d=\gcd(n, b-a)$. Then

• $H$ contains all the transpositions $(k \ k+\frac{n}{d})$

In particular, $H$ contains the subgroup generated by $(1 \ n/d+1 \ 2n/d+1\cdots )$, which is isomorphic to $S_d$, and $d!$ divises the order of $H$.

• $H$ contains all the transpositions $(1 \ 1+ \ell(b-a)),$

and that's pretty much it.

Thoughts.

I wonder if $H$ is isomorphic to a wreath product $S_d\wr S_{n/d}$ or something like that, which would solve the problem.

Of course, if one may show that $H$ is a maximal subgroup of $S_n$, then it would do the trick, since $H$ is transitive and imprimitive if $d>1$.

I woud be extremely surprised if the result was not already known.

Thanks for your help.