What new point group is formed if symmetry element i is added to C3 point group?


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  • $\begingroup$ Then it's not a group. $\endgroup$ – Zhe Feb 3 at 13:30
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    $\begingroup$ @Zhe the book answer says that S6 point group is formed $\endgroup$ – rishabhx64 Feb 3 at 15:39
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    $\begingroup$ A group plus one extra element is usually not a group at all. A group plus one extra element and the elements produced by it is usually another, bigger group (in this case $S_6$). $\endgroup$ – Ivan Neretin Feb 3 at 17:09
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    $\begingroup$ If it isn't obvious right away, we may have to write down all elements of the new group, one by one. $\endgroup$ – Ivan Neretin Feb 3 at 18:32
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    $\begingroup$ Forget the points, work with matrices. $\endgroup$ – Ivan Neretin Feb 3 at 19:13

Inversion $i$ is equivalent to a two-fold improper rotation $S_2$. Introducing a $2n$-fold improper rotation axis to the existing $C_n$ point group will result in a new $S_{2n}$ point group, e.g. in your case $C_3 ⊗ i = S_6$.

  • $\begingroup$ Is there a way to solve for the new group $\endgroup$ – rishabhx64 Feb 3 at 18:20
  • $\begingroup$ Can it be solved using a character table? $\endgroup$ – rishabhx64 Feb 3 at 18:20
  • $\begingroup$ @rishabhx64 Yes, but that would be an overkill for this case. $\endgroup$ – andselisk Feb 3 at 18:22
  • $\begingroup$ I'd say character tables are not for this purpose at all. $\endgroup$ – Ivan Neretin Feb 3 at 18:26
  • $\begingroup$ @andselisk take this case add $i$ to $C_{3v}$ $\endgroup$ – rishabhx64 Feb 3 at 18:26

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