I would say NCL or Carnival - preferably Caribbean. They tend to have a more youthful fun-loving, partying atmosphere/crowd. Mostly American.
I am not sure what they are like but maybe Island Cruises or Ocean Village too (more UK market perhaps) .
Yes I would suggest Carnival 'Fun Ships' for the type of atmosphere your looking for. I can't say the same about NCL as 'Freestyle Dining' for 20 people would be a nightmare trying to book tables to eat at the same time or even the same day! With 'fixed' dining you can prearrange 2 tables for 10, next to each other, at the time of booking & all enjoy each others company. Best of Luck & make sure you get a 'Group Discount'.
I myself, am not sure that 20 of you would want either the expense or trek involved in a caribbean cruise. Also hot weather and excessive drink (God forbid) do not make good bed partners! Carnival Cruise line advertise their fleet as "Fun Ships" and the Carnival Liberty is sailing from Dover all next summer, so maybe a look at these cruises might be worth while? At first glance, it may not seem a very good itinerary, as it only cruises to the Baltic capitals. However, with places like Tallin (hen and stag party capital of Europe) and Amsterdam in there, it may fit the bill? Plenty of opportunity for a bit of culture capture in places like Berlin and St.Petersburg also! The icing on the cake is the price of around £750 per person for a 12 day cruise at the moment! Another thought is Easy Cruise? They have just got themselves a new larger ship! That would involve flying down to the Med. and back though. Happy cruising.