# What is the symmetry of dxy orbital?

Considering the sign of orbital and assuming the $$z$$-axis as a principal axis, for me, it looks like that it has two perpendicular $$C_2$$ axes that penetrate the lobes, so I think it is $$C_\mathrm{2v}$$. But the book (Inorganic chemistry by Miessler and Tarr) says it is $$D_\mathrm{2h}$$.

Note that $$C_\mathrm{2v}$$ is a subgroup of $$D_\mathrm{2h}$$. This means that you're just overlooking some of the symmetry elements. In addition to the $$C_2$$ axes penetrating the lobes "end to end", you also have a $$C_2$$ axis perpendicular to the orbital plane through its center.
Your $$C_\mathrm{2v}$$ guess is also inconsistent because that point group only has one $$C_2$$ axis, but you said yourself that you already found two.
Imagine that the $$x$$ axis and $$y$$ axis lobes have a different color (phase). Locate the principal axis, that of highest rotational symmetry, if there is more than one just choose one. A $$C_2$$ or 180 degree rotation will make the orbital indistinguishable from its starting position if this points out of the plane of the orbital (i.e. I choose this to be along $$z$$ if orbital is in $$xy$$-plane). Any 'D' point group has a 2-fold, $$C_2$$ or 180 degree rotation axis, perpendicular to the principal axis so the $$\mathrm{d}_{xy}$$ belongs to the $$D_\mathrm{2h}$$ point group. You can check other symmetry elements by looking at the point group table. A $$C_\mathrm{2v}$$ molecule, i.e. one with symmetry of water molecule, has no 2-fold axis perpendicular to the principal axis. If you do not distinguish $$x$$ and $$y$$ with 'color' the orbital will have $$D_\mathrm{4h}$$ symmetry.