Some bismash products that are not group algebras II

We consider a family of Hopf algebras, each constructed as a bismash product from a factorisation of the symmetric group SN. These are semisimple as algebras over C and we show that they do not have the structure (as an algebra) of any group algebra if N⩾5. Previously, the author has established this in the special cases where N has the form p+1 or p+2 for p prime; here the general result is obtained by very different methods.