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Table 1 \(C_{3v}\) permutations

From: Automatic procedure for generating symmetry adapted wavefunctions

\({\hat{E}}\) \(({\hat{E}})\quad ({\hat{C}}_3)\quad ({\hat{C}}_3^2)\quad ({\hat{\sigma }})\quad ({\hat{\sigma }}')\quad ({\hat{\sigma }}'')\)
\({\hat{C}}_3\) \(({\hat{E}}\;{\hat{C}}_3\;{\hat{C}}_3^2)\quad ({\hat{\sigma }}\;{\hat{\sigma }}'\;{\hat{\sigma }}'')\)
\({\hat{C}}_3^2\) \(({\hat{E}}\;{\hat{C}}_3^2\;{\hat{C}}_3)\quad ({\hat{\sigma }}\;{\hat{\sigma }}''\;{\hat{\sigma }}')\)
\({\hat{\sigma }}\) \(({\hat{E}}\;{\hat{\sigma }})\quad ({\hat{C}}_3\;{\hat{\sigma }}'')\quad ({\hat{C}}_3^2\;{\hat{\sigma }}')\)
\({\hat{\sigma }}'\) \(({\hat{E}}\;{\hat{\sigma }}^{'})\quad ({\hat{C}}_3\;{\hat{\sigma }})\quad ({\hat{C}}_3^2\;{\hat{\sigma }}'')\)
\({\hat{\sigma }}''\) \(({\hat{E}}\;{\hat{\sigma }}'')\quad ({\hat{C}}_3\;{\hat{\sigma }}')\quad ({\hat{C}}_3^2\;{\hat{\sigma }})\)
  1. Permutation cycles of \(C_{3v}\)