Monomial bases for the primitive complex Shephard groups of rank three.

For the group algebra of each primitive non-Coxeter Shephard group of rank three, we construct a monomial basis and its explicit multiplication table. First, we find a Gröbner–Shirshov basis for the Shephard group of type L2. Then, since each of the groups of types L3 and M3 has a parabolic subgroup isomorphic to the group of type L2, by using the sets of minimal right coset representatives of L2 in L3 and M3, respectively, we apply the Gröbner–Shirshov basis technique to find the monomial bases for the Shephard groups of rank three L3 and M3. From this, we obtain the operation tables between the elements in each of the groups. Also we explicitly show that the group of type M3 has a subgroup of index 2 isomorphic to the group of type L3. [ABSTRACT FROM AUTHOR]