E. E. Floyd showed in 1973 that there exist only two nontrivial cobor-dism classes that contain manifolds with three cells, and that they lie in dimen-sions 10 and 5. We prove that there is an action of the cyclic group C2 on the 10-dimensional Floyd manifold which turns it into a conjugation manifold in the sense of Hausmann, Holm, and Puppe. The submanifold of fixed points is the 5-dimensional Floyd manifold, whose cohomology is isomorphic to that of the large one, scaled down by dividing the cohomological degree by a factor two.