  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.
