Permutations covered by subgroups?

Given integer $ m\in[1,n]$ fix a set $ \mathcal T$ of permutations in $ S_n$ . There are subgroups $ G_1,\dots,G_m$ of $ S_n$ so that $ \mathcal T$ is covered by cosets of $ G_1,\dots,G_m$ .

  1. My problem then is given $ \mathcal T$ is there always an $ m=O(poly(n))$ such that there are elements $ g_1,\dots,g_m\in S_n$ and some subgroups of $ G_1,\dots,G_m$ of $ S_n$ such that

$ $ \mathcal T\subseteq\cup_{i=1}^mg_iG_i$ $ $ $ (\sum_{i=1}^m|G_i|-|\mathcal T|)^2<m’$ $ where both $ m$ and $ m’$ are $ O(poly(n))$ .

  1. If not what is the trade off between $ m$ and $ m’$ ?

  2. Is it possible to get at least $ O(subexp(n))$ for both?

  3. If $ m’=0$ is there always a minimum $ m$ for all $ \mathcal T$ ?