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I'm reading David R. Finston and Patrick J. Morandi's book Abstract Algebra: Structure and Application and in its last section 10.6 "the 17 wallpaper groups" page 179 I'm confused on what it gives as an example of $pmg$:

enter image description here

According to its description on pg 178, $pmg$ has point group $D_2$ , so it shall contains a rotation. Wikipage on pmg also says so:

"The group pmg has two rotation centres of order two (180°), and reflections in only one direction. It has glide reflections whose axes are perpendicular to the reflection axes. The centres of rotation all lie on glide reflection axes."

But I couldn't find a rotation here in the pattern.

So, where did I miss?

Update for Doug’s answer:

If takes the red dot as the rotation centre, the two blue parts don’t match:

enter image description here

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  • $\begingroup$ @MarkBennet thx, good to know i didn't misunderstand it.. $\endgroup$ – athos Sep 27 '19 at 10:32
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    $\begingroup$ If the grey tiles crown up are all different from all the grey tiles crown down then no half turn can take any grey tile into another, so no half turn symmetries can exist. I reckon that is a good spot. (autocorrect typo corrected) $\endgroup$ – Mark Bennet Sep 27 '19 at 12:51
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You can rotate at the intersection points of $4$ tiles by $180$ degrees. There are two kinds of intersections points depending on whether say the upper right is white or grey.

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  • $\begingroup$ There are two kinds of grey tiles, pls see my question updated $\endgroup$ – athos Sep 27 '19 at 8:12
  • $\begingroup$ @athos You are correct on your picture but I'm confused when comparing to the pictures on wikipedia. The cell structure pictures do not seem to have any rotational symmetry. The picture labeled example and diagram for pmg does have the symmetry I described, as does the one labeled Egyptan tomb. So it seems that there are tilings with different amounts of symmetry being labeled as pmg? $\endgroup$ – quarague Sep 27 '19 at 8:33
  • $\begingroup$ I think it’s a drawing typo $\endgroup$ – athos Sep 27 '19 at 13:57
  • $\begingroup$ @athos That would be an explanation. I would prefer some other source to confirm but it seems the most likely. $\endgroup$ – quarague Sep 27 '19 at 14:23

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