Specifically, I am thinking of a cuboid with a given volume ($28\,000$) that has sides of integer length. For example, $20 \cdot 20 \cdot 70 = 28\,000$, but so do $10 \cdot 40 \cdot 70$ and $1 \cdot 1 \cdot 28\,000$. I am interested in finding how many possible integer combinations of side lengths there are that produce this volume.
Its prime factorisation is $2^5 \cdot 5^3 \cdot 7$, so I think that the answer may have something to do with permutations of those.
The order of the three groups does matter because there is a distinction between it being height, width or length.