A new way to represent products of Householder matrices is given that makes a typical Householder matrix algorithm rich in matrix-matrix multiplication. This is very desirable in that matrix-matrix multiplication is the operation of choice for an increasing number of important high performance computers. We tested the new representation by using it to compute the QR factorization on the FPS-164/MAX. Preliminary results indicate that it is a very efficient way to organize Householder computations.