  We have examined the synchronization and de-synchronization transitions
observable in the Kuramoto model with a standard pair-wise first harmonic
interaction plus a higher order (triadic) symmetric interaction for unimodal
and bimodal Gaussian distributions of the natural frequencies $\{ \omega_i \}$.
These transitions have been accurately characterized thanks to a
self-consistent mean-field approach joined with extensive numerical
simulations. The higher-order interactions favour the formation of two cluster
states, which emerge from the incoherent regime via continuous (discontinouos)
transitions for unimodal (bimodal) distributions. Fully synchronized initial
states give rise to two symmetric equally populated bimodal clusters, each
characterized by either positive or negative natural frequencies. These bimodal
clusters are formed at an angular distance $\gamma$, which increases for
decreasing pair-wise couplings until it reaches $\gamma=\pi$ (corresponding to
an anti-phase configuration), where the cluster state destabilizes via an
abrupt transition: the $\pi$-transition. The uniform clusters that reform
immediately after (with a smaller angle $\gamma$) are composed by oscillators
with positive and negative $\{ \omega_i \}$. For bimodal distributions we have
obtained detailed phase diagrams involving all the possible dynamical states in
terms of standard and novel order parameters. In particular, the clustering
order parameter, here introduced, appears quite suitable to characterize the
two cluster regime. As a general aspect, hysteretic (non hysteretic)
synchronization transitions, mostly mediated by the emergence of standing
waves, are observable for attractive (repulsive) higher-order interactions.
