A graphical group whose number of irreducible characters of degree p³ is not a polynomial in p

For a finite simple graph \(\Gamma\) and an odd prime \(p\), let \(G_\Gamma(\mathbb{F}_p)\) be the largest group of class at most two and exponent dividing \(p\) generated by elements \(x_v\), one for each vertex \(v\), such that \(x_u\) and \(x_v\) commute whenever \(u\) and \(v\) are not adjacent. Rossmann asked (Oberwolfach Report 38/2021; Question 1.10 in Bull. London Math. Soc. 2022) how the number of irreducible characters of \(G_\Gamma(\mathbb{F}_p)\) of degree \(p^i\) depends on \(p\), and suspected that these functions might be wild. We show that they are not always polynomials. Let \(\Gamma_{\mathrm{MK}}\) be the bipartite graph on the points and lines of the Möbius–Kantor configuration \(8_3\) in which a point is joined to the five lines not containing it. For every odd prime \(p\), \(G_{\Gamma_{\mathrm{MK}}}(\mathbb{F}_p)\) has exactly \(p^{10}\bigl(h(p)+\bigl(1+\bigl(\tfrac{-3}{p}\bigr)\bigr)(p-1)^{15}\bigr)\) irreducible characters of degree \(p^3\), where \(h\) is an explicit integer polynomial of degree 15 obtained by an exact computation; the count is a polynomial on each residue class modulo 3, but on no set of primes containing infinitely many primes \(\equiv 1\) and infinitely many \(\equiv 2 \pmod 3\), and this needs no computer. We use the description of character degrees by ranks due to O'Brien and Voll and the fact, classical over the complex numbers, that \(8_3\) can be realised exactly over the fields containing a root of \(t^2-t+1\). The counts are polynomials for \(i=1\), and for \(i\le 2\) if \(\Gamma\) is bipartite. Conditionally on the Mnëv–Sturmfels theorem as stated by Belkale and Brosnan, who sketch its proof, some count is not even a polynomial on residue classes (existence only). The general behaviour remains open. This is an unrefereed note.