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I am currently revising for module exam on groups and symmetry. I can easily identify rotations, reflections, translations and glide reflections that preserve the wallpaper group. However I am struggling to see which of these isometries generate the wallpaper group. For example wallpapergroup

I can see the translations , reflections, and glide reflections but the next question in the exam asks me to list a prove which isometries generate the wall paper group. How can I see/ know what generate the wallpaper group?

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There is a nice Patern Recognition Algorithm for identifying the wallpaper groups. It starts with the maximal rotation order, which can be $1,2,3,4,6$. The next question is, whether there is a mirror, or not; and so on. Once you know your wallpaper group, one can use a list of explicit generators of all $17$ groups, see here.

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