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Let $AGL(2d,2)$ be the affine general linear group acting natrually on a $2d$-dimensional vector space over $GF(2)$. Is there a regular subgroup of $AGL(2d,2)$ isomorphic to $Z_{2^d}:Z_{2^d}$ for $d=3$ and $4$ respectively?

I tried to compute all regular subgroups of a Sylow $2$-subgroup of $AGL(2d,2)$ using the magma command "RegularSubgroups" but failed since it's too memory-consuming.

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    @DerekHolt Is it because my version of magma v2.12-16 is too old? Did you tried to construct all the conjugacy classes of cyclic subgroup of order $8$ in $AGL(8,2)$?2012-11-23

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