Fano plane duality group ($G{L}_3(2)⋊⟨\tau ⟩,|\overline{G}|=336$)

The Fano plane duality group includes both incidence-preserving permutations of points and projective lines and the dualities that exchange those two types. Its type-preserving subgroup is $G{L}_3({\mathbb{F}}_2)$, of order $(8-1)(8-2)(8-4)=168$. A polarity supplies the inverse-transpose automorphism $A\mapsto (A^{-1}{)}^{\mathsf{T}}$, giving the semidirect productThere are $7\cdot 3=21$ unordered incident point-line pairs and $7\cdot 4=28$ unordered nonincident point-line pairs. The two transitive actions therefore have stabilizers of orders $336/21=16$ and $336/28=12$, respectively. These are dihedral groups. The normalizer of a Sylow subgroup of order seven has order42 and is $C_7⋊C_6$.

Every maximal proper subgroup is a conjugate subgroup of one of these three types or of the index-two collineation subgroup. The extension is also isomorphic to $PG{L}_2(7)$, identifying the geometric stabilizers with the standard finite-group classification.^[ref]