  We introduce a Hamiltonian lattice model for the $(1+1)$-dimensional
$\text{SU}(N_c)$ gauge theory coupled to one adjoint Majorana fermion of mass
$m$. The discretization of the continuum theory uses staggered Majorana
fermions. We analyze the symmetries of the lattice model and find lattice
analogs of the anomalies of the corresponding continuum theory. An important
role is played by the lattice translation by one lattice site, which in the
continuum limit involves a discrete axial transformation. On a lattice with
periodic boundary conditions, the Hilbert space breaks up into sectors labeled
by the $N_c$-ality $p=0, \ldots N_c-1$. Our symmetry analysis implies various
exact degeneracies in the spectrum of the lattice model. In particular, it
shows that, for $m=0$ and even $N_c$, the sectors $p$ and $p'$ are degenerate
if $|p-p'| = N_c/2$. In the $N_c = 2$ case, we explicitly construct the action
of the Hamiltonian on a basis of gauge-invariant states, and we perform both a
strong coupling expansion and exact diagonalization for lattices of up to $12$
lattice sites. Upon extrapolation of these results, we find good agreement with
the spectrum computed previously using discretized light-cone quantization. One
of our new results is the first numerical calculation of the fermion bilinear
condensate.
