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 $F_{21},$ the Frobenius group of order $21$. 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 orientation.

Moreover, since any triangular embedding of the complete graph $K_7$ into a surface is isomorphic, as is well known, to the classical (face $2$-colorable) toroidal biembedding, and since the two color classes define a pair of orthogonal Fano planes, we deduce, as an application of our previous result, that the group of the embedding automorphisms that preserve the color classes is the Frobenius group of order $21.$

In this way, we provide three geometrical representations of $F_{21}$. Also, we apply once more 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, thereby confirming again the potential of our Fano-plane approach.

Although some of the results in this paper may be (partially) known, we include direct and independent proofs in order to make the paper self-contained and offer a unified view on the subject.