For your case, I would ask hard questions to get a real sense of where the product org is strong/weak, so you can get a sense of (a) what you'd be walking into, and (b) where you would need to help the people reporting to you improve.
This sounds like you're really looking at stepping in as a product leader. I'd really want to know what are the dominant beliefs/practices in the culture, and how that sets everyone up, because culture change is brutal.
One example to start with I'd dig into is product vision — what is it? Do they have one? Is it inspiring, cohesive, unifying?
What about their day to day rhythms? Do they have proper discovery and delivery? Or is this huge org just doing product by committee and roadmaps, without actually setting their teams up to win?
Those are a few places I'd start. If you want to get a sense of some other questions to consider—and more importantly, the rationale behind them—have a scan of this article. I published it last week and it might be useful to you.
An epic resource for the topics you'd need to think about is EMPOWERED, Marty Cagan's latest book.