Past exam of the Probability, statistics and options II course of the University of Oxford / 2008 / A / 2 / b / iii / Solution

Relative to the three successive blocks of sizes $1,2,2$, the stabilizer consists of the block upper triangular matricesThe off-diagonal blocks contain $2+2+4=8$ arbitrary field entries. Thus its order is $(q-1)|\mathrm{GL}(2,q){|}^2{q}^8$. By orbit-stabilizer theorem,Equivalently, choose the line first and then a two-dimensional subspace of the four-dimensional quotient:This also checks the stabilizer count independently.