  Stimulated by the anomalous superconducting properties of UPt$_3$, we
investigate the pairing symmetry and the transition temperature in the
two-dimensional(2D) and three-dimensional(3D) hexagonal Hubbard model. We solve
the Eliashberg equation using the third order perturbation theory with respect
to the on-site repulsion $U$. As results of the 2D calculation, we obtain
distinct two types of stable spin-triplet pairing states. One is the
$f$-wave(B$_1$) pairing around $n = 1.2$ and in a small $U$ region, which is
caused by the ferromagnetic fluctuation. Then, the other is the $p_x$(or
$p_y$)-wave(E$_1$) pairing in large $U$ region far from the half-filling ($n =
1$) which is caused by the vertex corrections only. However, we find that the
former $f$-wave pairing is destroyed by introduced 3D dispersion. This is
because the 3D dispersion breaks the favorable structures for the $f$-wave
pairing such as the van Hove singularities and the small pocket structures.
Thus, we conclude that the ferromagnetic fluctuation mediated spin-triplet
state can not explain the superconductivity of UPt$_3$. We also study the case
of the pairing symmetry with a polar gap. This $p_z$-wave(A$_1$) is stabilized
by the large hopping integral along c-axis $t_z$. It is nearly degenerate with
the suppressed $p_x$(or $p_y$)-wave(E$_1$) in the best fitting parameter region
to UPt$_3$ ($1.3 \le t_z \le 1.5$). These two p-wave pairing states exist in
the region far from the half-filling, in which the vertex correction terms play
crucial roles like the case in Sr$_2$RuO$_4$.
