  Mutually unbiased bases (MUB) are interesting for various reasons. The most
attractive example of (a complete set of) MUB is the one constructed by
Ivanovi\'c as well as Wootters and Fields, which is referred to as the
canonical MUB. Nevertheless, little is known about anything that is unique to
this MUB. We show that the canonical MUB in any prime power dimension is
uniquely determined by an extremal orbit of the (restricted) Clifford group
except in dimension 3, in which case the orbit defines a special symmetric
informationally complete measurement (SIC), known as the Hesse SIC. Here the
extremal orbit is the one with the smallest number of pure states. Quite
surprisingly, this characterization does not rely on any concept that is
related to bases or unbiasedness. As a corollary, the canonical MUB is the
unique minimal 2-design covariant with respect to the Clifford group except in
dimension 3. In addition, these MUB provide an infinite family of highly
symmetric frames and positive-operator-valued measures (POVMs), which are of
independent interest.
