Genetic codes of specific low-rank groups

The article is devoted to the Coxeter groups of ranks ? 5 with generating set S, whose orders of pairwise products do not exceed 3, and to the search for natural additional relationships in which these groups are finite. A computer and specialized software “Magma” were used in the study. In the paper we present the defining relations of finite groups connected with the groups of low ranks ? 5 generated by 3-transpositions.