Orbit sizes and the central product group of order 16

Abstract

This paper continues the study of linear group actions with no regular orbits where the largest orbit size equals the order of the abelian quotient of the group. In previous work of the first author with Yong Yang it was shown that if G is a finite solvable group and G a finite group and V a finite faithful completely reducible G-module, possibly of mixed characteristic, and M is the largest orbit size in the action of G on V then $|G/G'|\le M$. In a continuation of this work the first author and his student Nathan Jones analyzed the first open case of when equality occurs and proved the following. If G is a finite nonabelian group and V a finite faithful irreducible G-module and $M=|G/G'|$ is the largest orbit of G on V and that there are exactly two orbits if size M on V, then $G=D_8$ and $V=V(2,3)$. This paper is concerned with the next case, the one where, under otherwise the same hypotheses as before, we have three orbits of size $M=|G/G'|$. It turns out that again there is exactly one such action, the one where G is the central product of $D_8$ and $C_4$ is acting on the vector space of order 25.