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 $Z\le {\mathbb{F}}_p^{\times }$ 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 Köhler 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.