From a native speaker - the possible answers you have been presented with (who, that, which) all make for very poor English. I appreciate this may have come from an English learning resource, but it has clearly not been created by a native speaker.
The only words that make sense in that gap would be:
His name was John and he belonged to a theatrical group.
Who, that and which are for introducing either a defining or non-defining clause - information that either confirms or adds to the preceding statement. The first statement is that 'his name was John', and the fact he went to a theatrical group says nothing at all about his name. His name and his hobby are just two unrelated facts about the man, and that is why they should be joined with 'and'.
You would use 'who' or 'that' in a construction like this:
I knew a man named John who belonged to a theatrical group.
The difference here is that the first statement is 'I knew a man', and the fact that he belonged to a theatrical group tells us more about that man you knew and is therefore a clause that helps us define them.