The Noether number of the non-abelian group of order 3p

It is proven that for any representation over a field of characteristic 0 of the non-abelian semidirect product of a cyclic group of prime order p and the group of order 3 the corresponding algebra of polynomial invariants is generated by elements of degree at most p+2. We also determine the exact degree bound for any separating system of the polynomial invariants of any representation of this group in characteristic not dividing 3p.Comment: 12 page

Groups with large Noether bound

The finite groups having an indecomposable polynomial invariant whose degree is at least half of the order of the group are classified. Apart from four sporadic exceptions these are exactly the groups having a cyclic subgroup of index at most two.Comment: Major revision: paper completely rewritten, results extended, title modified. Parts of the material moved to another preprint entitled "The Noether number for the groups with a cyclic subgroup of index two". To appear at Annales de l'Institut Fourie