Does trans-decalin really have a plane of symmetry?
It's not easy to see from a diagram, because it distorts bonds and angles. I recommend building it with a balls-and-sticks model set. You can also use a molecular viewer to model it; there are a couple of open-source (or at least free) ones out there.

I have calculated the molecule on the DF-BP86/def2-SVP level of theory. The point group of the molecule is C2h. In the following model I have highlighted the plane of symmetry, and the rotational C2-axis. At their intersection is an inversion centre Ci.



(I needed to downscale this a lot, click on the image to get to a high resolution still. Images created with ChemCraft, assembled with GIMP.)