I am interested in the permutation representation of the simple group $ G $ of order $ 168 $ given by $$ G=\langle (45)(67),(142)(356)\rangle $$ I think this permutation representation is $2$-transitive. For example Wikipedia says that viewing the simple group of order $168$ as $ GL_3(2)=PGL_3(2)=SL_3(2)=PSL_3(2) $ there is a natural $2$-transitive action on the $ 7 $ points of the Fano plane.
My question: Is the subgroup $ \langle(45)(67),(142)(356)\rangle $ of $ S_7 $ $2$-transitive?
Let's number the eight vectors in $ \mathbb{F}_2^3 $ by the numbers whose binary expansion they correspond to. So for example the standard basis is $$ 1\sim \begin{bmatrix} 1\\ 0 \\ 0 \\ \end{bmatrix}, 2 \sim \begin{bmatrix} 0\\ 1 \\ 0 \\ \end{bmatrix}, 4 \sim \begin{bmatrix} 0\\ 0 \\ 1 \\ \end{bmatrix}, $$ Then $ (45)(67) $ is the linear map $$ \tau=\begin{bmatrix} 1 & 0 & 1\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{bmatrix} $$ And $ (142)(356) $ is the linear map $$ \sigma=\begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\\ \end{bmatrix} $$ As @Jyrki Lahtonen points out it is basic linear algebra that $ GL_3(2) $ is $2$-transitive on the non zero vectors of $ \mathbb{F}_2^3 $ (since, again as he points out, any two distinct nonzero vectors over $ \mathbb{F}_2 $ are linearly independent). Since $ G $ is order $ 168 $ and $ GL_3(2) $ is order $ 168 $ then then $ \langle \tau,\sigma\rangle=GL_3(2) $ (this kind of assumes that the map $ G \to GL_3(2) $ is an isomorphism, or at least injective, which is maybe obvious, but not totally obvious to me?). So $ \langle\tau,\sigma\rangle $ is $2$-transitive and thus so is $ G $.