Aspects of the Linear Groups $GL(n,2),n<7$

Abstract

We provide tables which summarize various aspects of the finite linear groups $\text{GL}(n,2)$, $n<7$, in their action upon the vector space $V_n=V(n,2)$ and upon the associated projective space $\text{PG}(n–1,2)$. It is intended that the tabulated results should be immediately accessible to finite geometers, and to all others (design theorists, coding theorists, \ldots) who have occasional need of these groups. In the case $n=4$ attention is also paid to the maximal subgroup $\mathrm{\Gamma }\text{L}(2,4)$. In the case $n=6$ the maximal subgroups $\mathrm{\Gamma }\text{L}(2,8)$ and $\mathrm{\Gamma }\text{L}(3,4)$ are treated, as are class aspects of the tensor product structure $V_6=V_2\otimes V_3$, and of the exterior product structure $V_6={\wedge }^2{V}_4$.