Neither! Neither 'him + ING' nor 'him + infinitive' is correct. Of the six options you give, only the second option - "Ana suggested (that) Daniel pick group C" - is correct. Note that this is a subjunctive, which is why we use 'pick' rather than 'picks'. All the other options are wrong. Of these, the third one is the closest - you could say "Ana suggested to Daniel that he pick group C." This also involves a subjunctive clause. You can use the gerund following 'suggest' only when there is no addressee mentioned. For example 'Ana suggested picking group C'. This isn't possible if Daniel is part of the sentence - if you mention Daniel, it has to be a subjunctive clause. NB Also note that it's 'group C' as opposed to 'the group C'. The letter C makes this noun definite, so we don't use the definite article.