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I am reading an article and in one of the sections the article mentions the symmetry group.

The symmetry group of one of the objects the article talks about is the dihedral group of order 12, using this as an example the article talks about symmetry types denoted as : I,G,DD,R2,SD,D.

What is the meaning of the notation ? It's easy to see from the article that I is the id, but that's all I managed to figure out.

Edit: : link to the article - page 7 in the pdf, table 1 it is also said that "we define a symmetry type to be conjugacy class of subgroups of G"

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    It would help enormously if you indicated where this article could be found (ideally linked to it).2012-03-23
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    @QiaochuYuan - added2012-03-23
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    These are all defined in Table 1 on page 93 under the Effect column. $I$ does nothing, $G$ rotates $\pm\pi/3$, $DD$ rotates $\pm2\pi/3$, $R2$ by $\pi$, $SD$ are reflections through a coordinate plane and $D$ are reflections through a coordinate axis.2012-03-23
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    @bgins: No, Table I defines the operations; Table II defines the symmetry types.2012-03-23
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    Well together, they certainly define the meaning of the notation, dont't they? By the way, there is also another notation in [Conway's relatively new book](http://books.google.ch/books/about/The_symmetries_of_things.html?id=EtQCk0TNafsC&redir_esc=y). It would be nice to see the lattice of subgroups for these. I think the reason they define the symmetry type as a conjugacy class is because each symmetry should be invariant, for example, under the effect of rotations or reflections. But I think this is all made clearer when you view them in a lattice.2012-03-23
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    @bgins - that is the reason for the definition. but I still don't understand why conjugacy class...I thought this would be the quotient, after dividing by rotations&reflections. do you see why conjugacy class ?2012-03-23
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    Here's an intuitive explanation. Each group $G$ acts on itself by right or left multiplication. The conjugacy class of a particular element $a\in G$ tells you what the action of $a$ looks like under all "changes of basis", i.e., transform by any $g\in G$, then do $a$, then transform back with $g^{-1}$. It's certainly therefore very reasonable to want symmetry types to be invariant under conjugation.2012-03-23

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I don't think there's any meaning to the notation beyond what it's defined to denote in Table II right below Table I. If you have reason to believe that there's any additional meaning, please indicate it in the question.