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In $G=\operatorname{PGL}(4,5)$ there are two elementary abelian $2$-subgroups of order $16$ denoted by $E_{1}$ and $E_{2}$ with $N_{G}(E_{1})=E_{1}.\operatorname{Sp}(4,2)$ and $N_{G}(E_{2})=E_{2}.(2^{3}:S_{3})$.

$N_{G}(E_{1})$ is a maximal subgroup of $G$ and $(2^{3}:S_{3})$ is a point stabilizer of a nonidentity element of $E_{2}$ in $\operatorname{Sp}(4,2)$ (viewing $E_{2}$ as a symplectic basis of $\operatorname{Sp}(4,2)$).

By comparison, $H=\operatorname{PGL}(4,3)$ has $N_{H}(F_{1})=F_{1}.M_{1}$ and $N_{H}(F_{2})=F_{2}.M_{2}$ where $F_{i}$ are again elementary abelian $2$-subgroups of order $16$ and $M_{i}$ are, by some Magma computation, two maximal subgroups of $\operatorname{Sp}(4,2)$ of orders $72$ and $120$, respectively. I see that $2^{4}.\operatorname{Sp}(4,2)$ is not a subgroup of $H$ any more, in contrast. (It appears in the corresponding finite unitary group now.) Also $M_{1}$ and $M_{2}$ are stabilizers in $\operatorname{Sp}(4,2)$ of $f_{1}\in F_{1}$ and $f_{2}\in F_{2}$ which have orbit sizes $10$ and $6$.

Is there an explanation for this phenomenon? What are these orbits of sizes $10$ and $6$ now? I hope to generalizer this. For instance, in $S=\operatorname{PGL}(8,3)$, what are the analogous orbit sizes in $2^6$?

I've only just found out that $M_{1}\cong \operatorname{SO}^{+}_{4}(2)$ and $M_{2}\cong \operatorname{SO}^{-}_{4}(2)$. They do have orbit sizes 1,5,10 and 1,9,6….

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The difference between the examples arises principally because $5 \equiv 1 \bmod 4$ and $3 \equiv 3 \bmod 4$.

For $q \equiv 1 \bmod 4$, $G := {\rm GL}(4,q)$ has centre $Z$ divisible by $4$, and contains a group $S$ of symplectic type with $N_G(S) = ZS.{\rm Sp}(4,2)$. The group $S$ maps onto your group $E_1$ in ${\rm PGL}(4,q)$.

As you pointed out, you get similar behaviour in the unitary group when $q \equiv 3 \bmod 4$

When $q \equiv 3 \bmod 4$, $G := {\rm GL}(4,q)$ does not have an element of order $4$ in its centre. It has extraspecial subgroups $S^+$ and $S^-$ of two different types, with normalizers $ZS^+ {\rm GO}^+(4,2)$ and $ZS^- {\rm GO}^+(4,2)$ The groups $S^+$ and $S^-$ map onto your groups $F_1$ and $F_2$.

The inverse image of your group $E_2$ when $q=5$ has centre of order $4$ and derived group of order $2$ (as for $E_1$) but, unlike the inverse image of $E_1$, it has exponent $8$ and a smaller normalizer.

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  • $\begingroup$ Hello. Could you please point me to some reference including a proof of "When $q \equiv 3 \bmod 4$, $G := {\rm GL}(4,q)$ has extraspecial subgroups $S^+$ and $S^-$ of two different types, with normalizers $ZS^+ {\rm GO}^+(4,2)$ and $ZS^- {\rm GO}^+(4,2)$."? $\endgroup$
    – user488802
    Commented Dec 11, 2022 at 7:39
  • $\begingroup$ Also, do you mean"$ZS^+ {\rm GO}^+(4,2)$ and $ZS^- {\rm GO}^-(4,2)$ ."? Thank you. $\endgroup$
    – user488802
    Commented Dec 11, 2022 at 7:40
  • $\begingroup$ Yes, sorry, the second ${\rm GO}^+$ should be ${\rm GO}^-$. $\endgroup$
    – Derek Holt
    Commented Dec 11, 2022 at 8:07
  • $\begingroup$ Thank you. Do you have some reference for the first one? Appreciate your time! $\endgroup$
    – user488802
    Commented Dec 11, 2022 at 8:14
  • $\begingroup$ Sorry, but the only reference I have for this is Section 4.6 of "The Subgroup Structure of the Finite Classical groups" by Kleidman and Liebeck. The relevant information is contained in the last two lines of Table 4.6B, but you might struggle to track down a nice self-contained proof of these statements. $\endgroup$
    – Derek Holt
    Commented Dec 12, 2022 at 9:26

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