Classifying Triply-Invariant Subspaces

Let p be a prime number and consider the vector space consisting of all p-by-p-by-pmatrices with entries taken from the field with p elements. We wish to construct, list, and describe all those subspaces that are simultaneously invariant under three particular linear transformations on this vector space. Even for small primes p, this is an extensive and difficult computational problem. Using an elaborate overall strategy based on concepts from linear algebra, we completely solve this problem for the prime p=2, and we have completed several cases of this problem for the prime p=3. This problem has connections with classification problems for certain subgroups of wreath product finite groups of prime-power order.