  We study a class of twisted 3D $N=4$ supersymmetric Yang-Mills (SYM) theory
on particular 3-dimensional lattice denoted as $\mathcal{L}_{3D}^{su_3\times
u_1}$ and given by non trivial fibration $\mathcal{L}_{1D}^{u_1}\times
\mathcal{L}_{2D}^{su_3}$ with base $\mathcal{L}_{2D}^{su_3}=\mathbb{A}_2^\ast$,
the weight lattice of $SU(3)$. We first, develop the twisted 3D $N=4$ SYM in
continuum by using superspace method where the scalar supercharge $Q$ is
manifestly exhibited. Then, we show how to engineer the 3D lattice
$\mathcal{L}_{3D}^{su_3\times u_1}$ that host this theory. After that we build
the lattice action $\mathcal{S}_{latt}$ invariant under the 3 following: (i)
$U(N)$ gauge invariance, (ii) BRST symmetry, (iii) the hidden $SU(3) \times
U(1)$ symmetry of $\mathcal{L}_{3D}^{su_3\times u_1}$. Other features such as
reduction to twisted 2D supersymmetry with 8 supercharges living on
$\mathcal{L}_{2D}^{su_2\times u_1}$, the extension to twisted maximal 5D SYM
with 16 supercharges on lattice $\mathcal{L}_{5D}^{su_4\times u_1}$ as well as
the relation with known results are also given.
