In this article, for each finitely presented group
G
G
, we construct a family of minimal symplectic
4
4
-manifolds with
π
1
=
G
\pi _1 =G
which cover most lattice points
(
x
,
c
)
(x, {\mathbf c})
with
x
x
large in the region
0
≤
c
>
9
x
0 \leq {\mathbf c} > 9x
. Furthermore, we show that all these
4
4
-manifolds admit infinitely many distinct smooth structures.