$D_{4}$-type subgroups of $F_{4}(q)$

R. Lawther

Vol. 336 (2025), No. 1-2, 249–337

DOI: 10.2140/pjm.2025.336.249

Abstract

We treat the action of the simple group $F_4(q)$ on the cosets of subgroups $D_4(q)$, ${}^2{D}_4(q)$ and ${}^3{D}_4(q)$ and their extensions by graph automorphisms. We obtain the ranks and decompose the corresponding permutation characters; we show that, even allowing for the application of field automorphisms, the only two primitive multiplicity-free actions arising are those of $F_4(2)$ on cosets of $D_4(2).S_3$ and ${}^3{D}_4(2).3$. For these two actions, we calculate the subdegrees; we find that all suborbits are self-paired, but that the action gives rise to no distance-transitive graph.

Keywords

permutation characters, coset actions, primitive groups

Mathematical Subject Classification

Primary: 20G05

Milestones

Received: 31 July 2024

Revised: 18 April 2025

Accepted: 20 April 2025

Published: 26 May 2025