  We define the notion of $(p_0,p_1,\dots,p_d)$-type semi-equivelar gems for
closed connected PL $d$-manifolds, related to the regular embedding of gems
$\Gamma$ representing $M$ on a surface $S$ such that the face-cycles at all the
vertices of $\Gamma$ on $S$ are of the same type. The term is inspired by
semi-equivelar maps of surfaces. Given a surface $S$ having non-negative Euler
characteristic, we find all regular embedding types on $S$ and then construct a
genus-minimal semi-equivelar gem (if it exists) of each such type embedded on
$S$. Moreover, we present constructions of the following semi-equivelar gems:
  (1) For each closed connected surface $S$, we construct a genus-minimal
semi-equivelar gem that represents $S$. In particular, for $S=\#_n
(\mathbb{S}^1 \times \mathbb{S}^1)$ (resp., $\#_n(\mathbb{RP}^2)$), the
semi-equivelar gem of type $((4n+2)^3)$ (resp., $((2n+2)^3)$) is constructed.
  (2) For a closed connected orientable PL $d$-manifold $M$ (where $d \geq 3$)
of regular genus at most $1$, we show that $M$ admits a genus-minimal
semi-equivelar gem if and only if $M$ is a lens space.
  Moreover, if we consider semi-equivelar gems with $2$-gons then for a closed
connected orientable $d$-manifold $M$ (where $d \geq 3$) with
$\mathcal{G}(M)\leq 1$, $M$ admits a genus-minimal semi-equivelar gem (with
bigons).
