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Let $V$ be a finite vector space over the finite field $\mathbb F_q$.

Is there an algorithm to enumerate representatives for the double cosets $\mathrm{Aff}(V)\backslash S_V/\mathrm{Aff}(V)$, where $\mathrm{Aff}(V) = V\rtimes \mathrm{GL}(V)$ is the general affine group of the $\mathbb F_q$-vector space $V$ and $S_V$ is the symmetric group on the set $V$? If necessary you may assume $q=2$.

Alternatively a subgroup chain from $\mathrm{Aff}(V)$ to $S_V$ for applying the divide-and-conquer approach from section 4.6.8 of the Handbook of Computational Group Theory would be helpful.

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  • $\begingroup$ ${\rm Aff}(V)$ is generally maximal in the symmetric or alternating group on $V$, so the subgroup chain method won'r help much. $\endgroup$
    – Derek Holt
    Mar 5, 2015 at 20:48
  • $\begingroup$ @DerekHolt: Thank you for this info. I should have thought of O'Nan-Scott myself. $\endgroup$
    – j.p.
    Mar 6, 2015 at 15:48

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