If I may apply the native speakers' standards, [群]{ぐん} or [群]{む}れ looks really out of place with the other three words; It just does not belong in that particular group of words.
There is a sense of "officialness" associated with the words 組, 団 and 班 with membership, common purpose of existing, pecking order, etc. 群れ, however, refers to a group of people that just happen to be someplace at the same time. These people rarely form a social group.
Now the difficult part --- the differences among 組, 団 and 班. I am just giving real-life examples. There is a ton of exceptions, though, I warn you.
In schools (elementary thru high school), all students of each grade is divided into several 組(and those are called 一年三組、二年B組, etc.). If a teacher decides to divide his 組 into a number of small groups for a group project or something, those small groups are most often called 班.
A 班 is always a sub-group within a larger group. In some companies, a department may be divided into 班's.
団 is a group of people with the same or similar skills. It gives off a feeling of professionalism and often strong unity. Examples are circuses, orchestras, fire fighters, theatrical or ballet groups, etc.
Finally, for the curious, the generic name for an organized crime group is [暴力団]{ぼうりょくだん} but the actual individual groups are most often named ~~[組]{ぐみ} (not くみ). Some call themselves ~~会.