Abstract: The Lyness’s theory on construction high dimesional sysmetric integration rules is applied to four and six dimensional cases. The 9th and 11th degree symmetric quadrature formulae W⁽⁴⁾₅ and W⁽⁴⁾₆ for fourdimensional hypercubs, and the 9th and 11th degree symmetric quadrature formulae W^(6)*₅ and W^(6)*₆ for sixdimensional hypercubs is deduced. Compared to the same degree Gaussian quadrature formulae, these formulae need much less function evaluations. The number of function evaluations of W⁽⁴⁾₅ and W⁽⁴⁾₆ are 185 and 505, and the number of function evaluations of W^(6)*₅ and W^(6)*₆ are 465 and 1825 respectively. The deduced four and six dimensional formulae are used respectively in calculating impedance elements of RWG Galerkin moment method for arbitrary conducting surface objects and extensive RWG Galerkin moment method for inhomogeneous dielectric objects. The calculated results show that when the mesh cells’ electrical length is less than λ/4 , the quardrature relative precision of formulae W⁽⁴⁾₅ and W⁽⁴⁾₆are about 1e-6 for four dimensional, while that of formulae  W^(6)*₅ and W^(6)*₆are about 1e-4 for six dimensional.