Orbits of $Z \circ (2.O_8^+(2).2)$ in Dimension 8

Abstract

Groups of structure $2.O_8^+(2)$ have an irreducible representation of degree $8$ which can be realized over $\mathbb{Z}$ and any prime field $\mathbb{F}_p$. This representation extends to a group of structure $2.O_8^+(2).2$. Any subgroup $Z \leq \mathbb{F}_p^{\times}$ acts by scalar multiplication on this module over $\mathbb{F}_p$. In this short note we determine for which primes $p > 7$ and which $Z$ the central products $Z \circ (2.O_8^+(2)$ and $Z \circ (2.O_8^+(2).2)$ have a regular orbit on the $8$-dimensional $\mathbb{F}_p$-module. This work was triggered by an omission in the paper by K\"ohler and Pahlings with title 'Regular Orbits and the $k(GV)$-Problem', a paper which is used in various places in work on the $k(GV)$-problem.