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The maximal subgroups of the finite classical groups can be partitioned intonine classes according to Aschbacher's theorem. We are concerned here with theclass $\mathcal{C}_6$ consisting of normalisers of extraspecial groups or oftwo-groups of symplectic type. In "Constructing maximal subgroups of classicalgroups", Holt and Roney-Dougal give explicit constructions for generators ofthese subgroups. Namely, for $\mathrm{SL}(d, q)$, this is done by firstdescribing generators of the respective normalisers in $\mathrm{GL}(d, q)$ andafterwards appropriately scaling them so that they lie in $\mathrm{SL}(d, q)$.Effectively, this comes down to finding a $d$-th root of the determinants ofthe generators in $\mathbb{F}_q$.  In Proposition 9.5 of loc.cit., it is claimed that this can be done in almostall cases with only a few exceptions, but no detailed explanation or analysisis given why this is true or when exactly this is possible. This paper worksout the missing details by examining the relevant determinants and giving acomplete description of the cases in which they are $d$-th powers in$\mathbb{F}_q$. Furthermore, it turns out that these determinants are actuallyalready $1$ in many cases so that one can omit the scaling entirely -particularly for large $q$, this may save some time in practical computations.

$d$-th roots of determinants of generators of $\mathcal{C}_6$-subgroups of $\mathrm{GL}(d, q)$ in $\mathbb{F}_q$