Let $p$ be a prime, and $\Sigma\le S_{p-1}$ an abelian subgroup of order $p-1$ such that:
- all the elements of $\Sigma$ have cycle type $\underbrace{\left(\frac{p-1}{k},\dots,\frac{p-1}{k}\right)}_{k\text{ slots}}$, for some divisor $k$ of $p-1$ depending on the element;
- $\Sigma$ has a subgroup $T\cong C_q\times C_q$, for some prime $q=\frac{p-1}{m}$ (therefore, every nontrivial $\tau\in T$ has cycle type $\underbrace{\left(q,\dots,q\right)}_{m\text{ slots}}$).
I'm looking for the explicit form (list of elements) of some $\Sigma\cong C_2\times C_2\times C_3\le S_{12}$ (case $p=13$) and $\Sigma\cong C_3\times C_3\times C_2\le S_{18}$ (case $p=19$). I add GAP and Magma tags in case these c.a.s. may assist.