The separating Noether number of small groups

Abstract: The separating Noether number of a finite group is the minimal positive integer $d$ such that for any finite dimensional complex linear representation of the group, any two dictinct orbits can be distinguished by a polynomial invariant of degree at most $d$. In the present paper its exact value is determined for all groups of order less than $32$, for all finite groups with a cyclic subgroup of index at most $2$, and for the direct product of a dihedral group with the $2$-element group. The results show in particular that unlike the ordinary Noether number, the separating Noether number of a non-abelian finite group may well be equal to the separating Noether number of a proper direct factor of the group. Most of the results are proved for the case of a general (possibly finite) base field containing an element whose multiplicative order equals the size of the group.