  Motivated by a recent angle-resolved thermal conductivity experiment that
shows a twofold gap symmetry in the high-field and low-temperature C phase in
the heavy-fermion superconductor UPt$_3$, we group-theoretically identify the
pairing functions as $E_{1u}$ with the $f$-wave character for all the three
phases. The pairing functions are consistent with the observation as well as
with a variety of existing measurements. By using a microscopic quasi-classical
Eilenberger equation with the identified triplet pairing function under applied
fields, we performed detailed studies of the vortex structures for three
phases, including the vortex lattice symmetry, the local density of states, and
the internal field distribution. These quantities are directly measurable
experimentally by SANS, STM/STS, and NMR, respectively. It is found that, in
the B phase of low $H$ and low $T$, the double-core vortex is stabilized over a
singular vortex. In the C phase, thermal conductivity data are analyzed to
confirm the gap structure proposed. We also give detailed comparisons of
various proposed pair functions, concluding that the present scenario of
$E_{1u}$ with the $f$-wave, which is an analogue to the triplet planar state,
is better than the $E_{2u}$ or $E_{1g}$ scenario. Finally, we discuss the
surface topological aspects of Majorana modes associated with the $E_{1u}^f$
state of planar like features.
