For the group algebra of the finite non-crystallographic Coxeter group of type H4, its Gröbner-Shirshov basis is constructed as well as the corresponding standard monomials, which describe explicitly all symmetries acting on the 120-cell and produce a natural operation table between the 14400 elements for the group.
Keywords: Coxeter groups; Gröbner–Shirshov basis; standard monomials.