  We discuss Type IIB 5-brane configurations for 5d $\mathcal{N}=1$ gauge
theories with hypermultiplets in the rank-3 antisymmetric representation and
with various other hypermultiplets, which flow to a UV fixed point at the
infinite coupling. We propose 5-brane web diagrams for the theories of $SU(6)$
and $Sp(3)$ gauge groups with rank-3 antisymmetric matter and check our
proposed 5-brane webs against several consistency conditions implied from the
one-loop corrected prepotential. Using the obtained 5-brane webs for rank-3
antisymmetric matter, we apply the topological vertex method to compute the
partition function for one of these $SU(6)$ gauge theories.
