A class of multifactorial designs are defined and analyzed. The designs considered have each a total number of observations that can not be divided equally among the cells of the designs; however, by distributing the observations in a way that is in a certain sense symmetrical, the equations that determine the least squares estimates of the linear parameters become explicitly solvable. The case of two non-interacting factors with arbitrary numbers of levels is treated first. In the $n$-factor case we have to restrict ourselves to factors having equal numbers of levels. After defining the designs, the estimates are computed. Some general discussions of the symmetries and algebraic properties involved conclude the paper.