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I am new to group theory so please forgive me for my stupid question. Why are the groups $C_2\times C_3$ not the same as $D_6$? Aren't they both generated by 2 elements one of order 2 the other 3? Is it because for the former the elements are ordered pairs but the latter isn't? And $(a,b)$ in the former means there isn't a $(b,a)$ whereas for $D_6$ we can have $ab$ and $ba$?

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    Does $D_6$ have order 6 or 12? See http://en.wikipedia.org/wiki/Dihedral_group#Notation.2012-02-06

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