Direct Product Tables

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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 \(D_{4h}\)
	\(A_1\)	\(A_2\)	\(B_1\)	\(B_2\)	\(E\)
\(A_1\)	\(A_1\)	\(A_2\)	\(B_1\)	\(B_2\)	\(E\)
\(A_2\)	-	\(A_1\)	\(B_2\)	\(B_1\)	\(E\)
\(B_1\)	-	-	\(A_1\)	\(A_2\)	\(E\)
\(B_2\)	-	-	-	\(A_1\)	\(E\)
\(E\)	-	-	-	-	\(A_1 + [A_2] + B_1 + B_2\)