You can see the way that the p4g symmetry group works by watching how the animation builds the tiling starting with the square tile shown on the left. You rotate about the lower left corner and reflect across the red lines. If you think about how these symmetries act on the entire tiling, you can see that if you reflect over the red lines, or rotate 90 degrees about any of the yellow points, the new tiling matches up with the original perfectly. Therefore, we say the tiling as a while possesses reflexive and four-fold symmetries.

To see that p4g tilings are also symmetric under translations, imagine rotating and reflecting four copies of the original tile to obtain the square to the right. You can then wallpaper the plane with this larger square using only translations. After some thought it is not difficult to see that this new wallpaper is the same as the original tiling. This is just what we means when we say a tiling is symmetric under translation. Sometimes such tilings are called periodic.

Kali denotes this symmetry by "4*2". You can go to now to experiment with symmetry p4g.