  We propose an exactly solvable lattice Hamiltonian model of topological
phases in $3+1$ dimensions, based on a generic finite group $G$ and a
$4$-cocycle $\omega$ over $G$. We show that our model has topologically
protected degenerate ground states and obtain the formula of its ground state
degeneracy on the $3$-torus. In particular, the ground state spectrum implies
the existence of purely three-dimensional looplike quasi-excitations specified
by two nontrivial flux indices and one charge index. We also construct other
nontrivial topological observables of the model, namely the $SL(3,\mathbb{Z})$
generators as the modular $S$ and $T$ matrices of the ground states, which
yield a set of topological quantum numbers classified by $\omega$ and
quantities derived from $\omega$. Our model fulfills a Hamiltonian extension of
the $3+1$-dimensional Dijkgraaf-Witten topological gauge theory with a gauge
group $G$. This work is presented to be accessible for a wide range of
physicists and mathematicians.
