  We present a method for computing the table of marks of a direct product of
finite groups. In contrast to the character table of a direct product of two
finite groups, its table of marks is not simply the Kronecker product of the
tables of marks of the two groups. Based on a decomposition of the inclusion
order on the subgroup lattice of a direct product as a relation product of
three smaller partial orders, we describe the table of marks of the direct
product essentially as a matrix product of three class incidence matrices. Each
of these matrices is in turn described as a sparse block diagonal matrix. As an
application, we use a variant of this matrix product to construct a ghost ring
and a mark homomorphism for the rational double Burnside algebra of the
symmetric group~$S_3$.
