We compute the class of the compactified Hurwitz
divisor $\overline{\mathfrak{TR}}_d$ in $\overline{\mathcal{M}}_{2d-3}$ consisting of curves of genus
$g=2d-3$ having a pencil $\mathfrak g^1_d$ with two unspecified
triple ramification points. This is the first explicit example of a
geometric divisor on $\overline{\mathcal{M}}_g$ which is not pulled-back form the
moduli space of pseudo-stable curves. We show that the intersection
of $\overline{\mathfrak{TR}}_d$ with the boundary divisor $\Delta_1$ in $\overline{\mathcal{M}}_g$ picks-up
the locus of Fermat cubic tails.