Presentations Of The Imprimitive Complex Reflection Groups G(m,p,n)

Prof. Jian-yi Shi introduced an essential presentation and a congruent relation among presentations for a complex reflection group. He established a bijection between the set of all the congruence classes of presentations for the group G(m, 1,n) and the set of isomorphism classes of all the rooted trees of n nodes. He also established a bijection between the set of all the congruence classes of presentations for the group G(m, m, n) and the set of isomorphism classes of all the connected graphs with n nodes and n edges. He still gave an explicit description in terms of rooted graphs for representatives of all the congruence classes of presentations for the imprimitive complex reflection group G(m,p,n) in his paper [1][2].In this paper ,we make a simplification for all the presentations of the group G(m, m, n) and G(m,p, n). Then we give all the non-congruent essential presentations of the imprimitive complex reflection groups G(m, m, n)(when m=3,n=3,4,5,6),G(10,2,3) and G(6,3,4).