  We study the Tavis-Cummings model with three modes of oscillation by using
four different algebraic methods: the Bogoliubov transformation, the
normal-mode operators, and the tilting transformation of the $SU(1,1)$ and
$SU(2)$ groups. The algebraic method based on the Bogoliubov transformation and
the normal-mode operators let us obtain the energy spectrum and eigenfunctions
of a particular case of the Tavis-Cummings model, while with the tilting
transformation we are able to solve the most general case of this Hamiltonian.
Finally, we compute some expectation values of this problem by means of the
$SU(1,1)$ and $SU(2)$ group theory.
