  The finite Heisenberg group knows when the dimension of Hilbert space is a
square number. Remarkably, it then admits a representation such that the entire
Clifford group --- the automorphism group of the Heisenberg group --- is
represented by monomial phase-permutation matrices. This has a beneficial
influence on the amount of calculation that must be done to find Symmetric
Informationally Complete POVMs. I make some comments on the equations obeyed by
the absolute values of the components of the SIC vectors, and on the fact that
the representation partly suggests a preferred tensor product structure.
