A contribution to the analysis of a reduction algorithm for groups with an extraspecial normal subgroup

Abstract

Reduction algorithms are an important tool for understanding structural properties of groups. They play an important role in algorithms designed to investigate matrix groups over a finite field. One such algorithm was designed by Brooksbank et al. for members of the class $C_6$ in Aschbacher's theorem, namely groups $N$ that are normalizers in $GL(d,q)$ of certain absolutely irreducible symplectic-type $r$-groups $R$, where $r$ is a prime and $d=r^n$ with $n>2$. However, the analysis of this algorithm has only been completed when $d=r^2$ and when $d=r^n$ and $n>2$, in the latter case under the condition that $G/RZ(G)\cong N/RZ(N)$. We prove that the algorithm runs successfully for some groups in the case of $d=r^3$ without any assumption.