  The "canonical dimension" of an algebraic group over a field by definition is
the maximum of the canonical dimensions of principal homogenous spaces under
that group. Over a field of characteristic zero, we prove that the canonical
dimension of the projective linear group PGL_6 is 3. We give two distinct
proofs, both of which rely on the birational classification of rational
surfaces over a nonclosed field. One of the proofs involves taking a novel look
at del Pezzo surfaces of degree 6.
