Orthogonal and oriented Fano planes, triangular embeddings of $K_7,$ and geometrical representations of the Frobenius group $F_{21}$

In this paper we present some geometrical representations of the Frobenius group of order $21$ (henceforth, $F_{21}$). The main focus is on investigating the group of common automorphisms of two orthogonal Fano planes and the automorphism group of a suitably oriented Fano plane. We show that both groups are isomorphic to $F_{21},$ independently of the choice of the two orthogonal Fano planes and of the choice of the orientation. We show, moreover, that any triangular embedding of the complete graph $K_7$ into a surface is isomorphic to the classical toroidal biembedding and hence is face $2$-colorable, with the two color classes defining a pair of orthogonal Fano planes. As a consequence, we show that, for any triangular embedding of $K_7$ into a surface, the group of the automorphisms that preserve the color classes is the Frobenius group of order $21.$ This way we provide three geometrical representations of $F_{21}$. Also, we apply the representation in terms of two orthogonal Fano planes to give an alternative proof that $F_{21}$ is the automorphism group of the Kirkman triple system of order $15$ that is usually denoted as #61.