Didn’t Heinlein or Clark write a book about it....”The Ark” or something like it.
Heinlein’s Orphans of the Sky was a generation starship novel, as was Aldis’s Starship (kinda dumb US title, revealing the big secret in the title). And then there was the TV show The Starlost.
I liked the Space: 1999 episode with Joan Collins on the generation starship that suffered an accident and the survivors turned to cannibalism to survive.
I don’t know but there would not be enough genetic diversity there without continually sending more and more people.