  We study the quantum mechanics of 3-index Majorana fermions $\psi^{abc}$
governed by a quartic Hamiltonian with $O(N)^3$ symmetry. Similarly to the
Sachdev-Ye-Kitaev model, this tensor model has a solvable large $N$ limit
dominated by the melonic diagrams. For $N=4$ the total number of states is
$2^{32}$, but they naturally break up into distinct sectors according to the
charges under the $U(1)\times U(1)$ Cartan subgroup of one of the $O(4)$
groups. The biggest sector has vanishing charges and contains over $165$
million states. Using a Lanczos algorithm, we determine the spectrum of the
low-lying states in this and other sectors. We find that the absolute ground
state is non-degenerate. If the $SO(4)^3$ symmetry is gauged, it is known from
earlier work that the model has $36$ states and a residual discrete symmetry.
We study the discrete symmetry group in detail; it gives rise to degeneracies
of some of the gauge singlet energies. We find all the gauge singlet energies
numerically and use the results to propose exact analytic expressions for them.
