  We study the $O(N_1)\times O(N_2)\times O(N_3)$ symmetric quantum mechanics
of 3-index Majorana fermions. When the ranks $N_i$ are all equal, this model
has a large $N$ limit which is dominated by the melonic Feynman diagrams. We
derive an integral formula which computes the number of $SO(N_1)\times
SO(N_2)\times SO(N_3)$ invariant states for any set of $N_i$. For equal ranks
the number of singlets is non-vanishing only when $N$ is even, and it exhibits
rapid growth: it jumps from $36$ in the $O(4)^3$ model to $595354780$ in the
$O(6)^3$ model. We derive bounds on the values of energy, which show that they
scale at most as $N^3$ in the large $N$ limit, in agreement with expectations.
We also show that the splitting between the lowest singlet and non-singlet
states is of order $1/N$. For $N_3=1$ the tensor model reduces to $O(N_1)\times
O(N_2)$ fermionic matrix quantum mechanics, and we find a simple expression for
the Hamiltonian in terms of the quadratic Casimir operators of the symmetry
group. A similar expression is derived for the complex matrix model with
$SU(N_1)\times SU(N_2)\times U(1)$ symmetry. Finally, we study the $N_3=2$ case
of the tensor model, which gives a more intricate complex matrix model whose
symmetry is only $O(N_1)\times O(N_2)\times U(1)$. All energies are again
integers in appropriate units, and we derive a concise formula for the
spectrum. The fermionic matrix models we studied possess standard 't Hooft
large $N$ limits where the ground state energies are of order $N^2$, while the
energy gaps are of order $1$.
