A Negative Answer to a Question of Praeger and Seress on Minimal Subdegrees of Primitive Groups of Twisted Wreath Type

Let \(G\) be a finite primitive permutation group of twisted wreath type, with socle \(T^k\) and point stabiliser \(P\), and let \(H\) be its primitive component: the group induced on one coordinate of the product decomposition \(\Delta^k\), where \(|\Delta|=|T|\). Praeger and Seress (Oberwolfach Report 12/2005, on joint work with Giudici, Li and Trofimov) state that the least non-trivial subdegree satisfies \(\operatorname{MinSubDeg}(G)\le k\cdot\operatorname{MinSubDeg}(H)\), and they ask in their Question 4 (p. 692) whether this inequality is always strict. We give a negative answer to Question 4. Let \(P=\operatorname{PGL}(3,4)\) act on the \(21\) points of the projective plane of order \(4\), let \(Q\) be the stabiliser of a point \(y\), let \(T=\operatorname{PSL}(2,4)\cong A_5\), and let \(Q\) act on \(T\) through its action on the quotient space \(\mathbb{F}_4^3/y\). The twisted wreath product \(G=T\operatorname{twr} P\), acting on its base group, is primitive of twisted wreath type of degree \(60^{21}\), with \(k=21\), \(H=A_5\times A_5\), \(\operatorname{MinSubDeg}(H)=12\) and \(\operatorname{MinSubDeg}(G)=252=21\cdot12\). By a reduction which is known in substance, and which we prove again, the last equality is a statement about the subgroups of \(\operatorname{PGL}(3,4)\) of order greater than \(240\). We prove this statement in two ways: computer-assisted, by exhaustive exact computations with independently written programs; and by a written argument from the list of the maximal subgroups of \(\operatorname{PSL}(3,4)\) and \(\operatorname{PGL}(3,4)\) in the tables of Connor and Leemans. Computer-assisted, we also find equality for the top groups \(\operatorname{P\Gamma L}(3,4)\) \((315=21\cdot15)\) and \(\operatorname{PSL}(3,5)\) \((465=31\cdot15)\), where \(H\) is the full holomorph of \(A_5\), and strict inequality for \(\operatorname{PSL}(3,4)\) and \(\operatorname{P\Sigma L}(3,4)\). The full paper of Giudici, Li, Praeger, Seress and Trofimov (2006) and its companion paper were not accessible to us; we make no statement that depends on their content. No answer to Question 4 was found in the literature accessible to us; a search that finds nothing is not a proof of novelty. We claim neither that the example is the smallest one nor a classification of the cases of equality, and the question remains open under additional hypotheses. This is an unrefereed note.