  We study D-modules and related invariants on the space of 2 x 2 x n
hypermatrices for n >= 3, which has finitely many orbits under the action of G
= GL_2 x GL_2 x GL_n. We describe the category of coherent G-equivariant
D-modules as the category of representations of a quiver with relations. We
classify the simple equivariant D-modules, determine their characteristic
cycles and find special representations that appear in their G-structures. We
determine the explicit 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 and n=4, they are completely
uniform for n >= 5.
