Algebraic foundation of maximal finite group of 3 dimentional cryatallography

Four arithmetic non-equivalent metric tenser matrices, T-1 = (0(0)(1) 1(0)( 0) 0(1)(0)), T-2 = (1(0)(2) 2(0)(1) 0(2)(0)), T-3 = (1(1)(2) 2(1)(1) 1(2)(1 )) and T-4 = (-1(-1)(3) 3(-1)(-1) -1(3)(-1)) have been derived in this pape r according to a crystallographic general equation (N) over tilde TN> = T. T-1, T-3 and T-4 are geometric equivalent ones, therefore, only T-1 and T-2 are geometric non-equivalent ones.Substituting T-1 and T-2 into (N) over t ilde TN = T, two maximal finite groups can be derived, which have 48 and 24 elements respectively and belong to two crystallographic point groups. The other 30 point groups can be derived according to group-subgroup relations hip.