  We express Alltop's construction of mutually unbiased bases as orbits under
the Weyl-Heisenberg group in prime dimensions and find a related construction
in dimensions 2 and 4. We reproduce Alltop's mutually unbiased bases using
abelian subgroups of the Clifford group in prime dimensions, in direct analogy
to the well-known construction of mutually unbiased bases using abelian
subgroups of the Weyl-Heisenberg group. Finally, we prove three theorems
relating to the distances and linear dependencies among different sets of
mutually unbiased bases.
