We study
D
\mathcal {D}
-modules and related invariants on the space of
2
×
2
×
n
2\times 2\times n
hypermatrices for
n
≥
3
n\geq 3
, which has finitely many orbits under the action of
G
=
G
L
2
(
C
)
×
G
L
2
(
C
)
×
G
L
n
(
C
)
G=GL_2(\mathbb {C}) \times GL_2(\mathbb {C}) \times GL_n(\mathbb {C})
. We describe the category of coherent
G
G
-equivariant
D
\mathcal {D}
-modules as the category of representations of a quiver with relations. We classify the simple equivariant
D
\mathcal {D}
-modules, determine their characteristic cycles and find special representations that appear in their
G
G
-structures. We determine the explicit
D
\mathcal {D}
-module structure of the local cohomology groups with supports given by orbit closures. As a consequence, we calculate the Lyubeznik numbers and intersection cohomology groups of the orbit closures. All but one of the orbit closures have rational singularities: we use local cohomology to prove that the one exception is neither normal nor Cohen–Macaulay. While our results display special behavior in the cases
n
=
3
n=3
and
n
=
4
n=4
, they are completely uniform for
n
≥
5
n\geq 5
.