Do all unitary simple groups $U_{2n+1}(2)$ have maximal subgroups of the form $3^{2n}:S_{2n+1}$?

Question

In the ATLAS, the unitary simple groups $U_5(2)$ and $U_7(2)$ have maximal subgroups of structures $3^4:S_5$ and $3^6:S_7$, respectively. It seems that they are subgroups of the generalized symmetric groups $3^5:S_5$ and $3^7:S_7$. Do all the unitary simple groups $U_{2n+1}(2)$ have maximal subgroups of the form $3^{2n}:S_{2n+1}$? If so, where can one obtain suitable reading material on it?

Answer 1

Yes that's almost correct, but it is ${\rm SU}(n,2)$ that has a maximal subgroup with that structure. You have to divide the order by $\gcd(3,n)$ for its image in $U_n(2) = {\rm PSU}(n,2)$.

More generally ${\rm SU}(n,q)$ has a maximal imprimitive subgroup with the structure $(q+1)^{n-1}:S_n$, and again you divide by $\gcd(q+1,n)$ for the order of its image in ${\rm PSU}(n,q)$.

There are a few small exceptions when the subgroup is not maximal: $(n,q)=(3,5), (4,3), (6,2)$.

For information on maximal subgroups of classical groups see Kleidman & Liebeck "The Subgroup Structure of the Finite Classical Groups" (Table 3.5B for the unitary case) for dimensions greater than $12$, or the tables in Bray, Holt & Roney-Dougal "The Maximal Subgroups of the Low-Dimensional Finite Classical Groups" for dimensions up to $12$.