Skip to main content
Chemistry LibreTexts

Direct Product Tables

  1. Last updated
  2. Save as PDF
  • Page ID
    54310

    • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

    Template:hideTOC

    Table 1: For \(C_2\), \(C_{2v}\), \(C_{2 h}\); \(C_3\), \(C_{3 v}\), \(C_{3 h}\); \(D_3\), \(D_{3 d}\),\( D_{3 h}\) ; \(C_6\), \(C_{6 v}\), \(C_{ 6 h}\); \(D_6\), and \(S_{6}\)
    \(A_1\) \(A_2\) \(B_1\) \(B_2\) \(E_1\) \(E_2\)
    \(A_1\) \(A_1\) \(A_2\) \(B_1\) \(B_2\) \(E_1\) \(E_2\)
    \(A_2\) - \(A_1\) \(B_2\) \(B_1\) \(E_1\) \(E_2\)
    \(B_1\) - - \(A_1\) \(A_2\) \(E_2\) \(E_1\)
    \(B_2\) - - - \(A_1\) \(E_2\) \(E_1\)
    \(E_1\) - - - - \(A_1 + [A_2] + E_2\) \(B_1+B_2+E_1\)
    \(E_2\) - - - - - \( A_1 + [A_2] +E_2\)
    Table 2: For \(T\), \(T_{h}\), \(T_{d}\); \(O\), and \(O_{h}\)
    \(A\) \(B_1\) \(B_2\) \(T\) \(T_2\)
    \(A_1\) \(A_1\) \(A_2\) \(E\) \(T_1\) \(T_2\)
    \(A_2\) - \(A_1\) \(E\) \(T_2\) \(T_1\)
    \(E\) - - \(A_1 + [A_2] + E_2\) \(T_1 + T_2\) \(T_1+T_2\)
    \(T_1\) - - - \(A_1 + E+ [T_1] + T_2\) \(A_2+E + T_1+T_2\)
    \(T_2\) - - - - \(A_1+E +[T_1] +T_2\)
    Table 3: For \(C_{\infty v}\), and \(D_{\infty h}\)
    \(\Sigma^+\) \(\Sigma^-\) \(\Pi\) \(\Delta\) -
    \(\Sigma^+\) \(\Sigma^+\) \(A_2\) \(\Pi\) \(\Delta\) -
    \(\Sigma^-\) - \(\Sigma^+\) \(\Pi\) \(\Delta\) -
    \(\Pi\) - - \( \Sigma^+ + [ \Sigma^- ] + \Delta \) \(\Delta + \Phi\) -
    \(\Delta\) - - - \( \Sigma^+ + [ \Sigma^- ] + \Sigma \) -
    - - - - - -
    Table 4: For \(D_{2}\), and \(D_{2h}\)
    \(A\) \(B_1\) \(B_2\) \(B_3\)
    \(A_1\) \(A\) \(B_1\) \(B_2\) \(B_3\)
    \(B_1\) - \(A\) \(B_3\) \(B_2\)
    \(B_2\) - - \(A\) \(B_1\)
    \(B_3\) - - - \(A\)
    Table 5: For \(C_{6v}\), \(D_6\) and \(D_{6h}\)
    \(A_1\) \(A_2\) \(B_1\) \(B_2\) \(E_1\) \(E_2\)
    \(A_1\) \(A_1\) \(A_2\) \(B_1\) \(B_2\) \(E_1\) \(E_2\)
    \(A_2\) - \(A_1\) \(B_2\) \(B_1\) \(E_1\) \(E_2\)
    \(B_1\) - - \(A_1\) \(A_2\) \(E_2\) \(E_1\)
    \(B_2\) - - - \(A_1\) \(E_2\) \(E_1\)
    \(E_1\) - - - - \(A_1 + [A_2] + E_2\) \(B_1 + [B_2] + E_1\)
    \(E_2\) - - - - - \(A_1 + [A_2] + E_2\)
    Table 6: For \(C_{4v}\), \(D_4\), \(D_{2d}\), and \(D_{4h}\)
    \(A_1\) \(A_2\) \(B_1\) \(B_2\) \(E\)
    \(A_1\) \(A_1\) \(A_2\) \(B_1\) \(B_2\) \(E\)
    \(A_2\) - \(A_1\) \(B_2\) \(B_1\) \(E\)
    \(B_1\) - - \(A_1\) \(A_2\) \(E\)
    \(B_2\) - - - \(A_1\) \(E\)
    \(E\) - - - - \(A_1 + [A_2] + B_1 + B_2\)

    Direct Product Tables is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

    1. Back to top
    • Was this article helpful?

    Recommended articles

    1. Article type
      Section or Page
      License
      CC BY-NC-SA
      License Version
      4.0
      Show Page TOC
      no on page
    2. Tags
      This page has no tags.