- The industry: You say this is a new industry for you. Every industry has characteristics that are unexpected for an outsider. Until you know something about this one, you won't have anything useful to say to those around you, because, as they'll remind you, "This industry is different," even if it isn't.
- The company: Different companies have different dynamics -- cultures, habits, manners of speech and dress, and ways of going about things. You don't want to be the corporate equivalent of the tourist who figures the locals will understand what he's saying if only he speaks louder and more slowly. You need to get a feel for how business is done around here.
- The individuals: Specifically, you need to learn who matters, what each expects, what each one wants, whom each one trusts and distrusts, and how each one thinks and makes decisions. Businesses are networks of relationships before they are collections of processes. You need to start building yours -- with those above you in the hierarchy; with your peers; with those your organization interacts with; and with the people who report to you.
Spend as much time as you can meeting with people one-on-one. Ask them what they expect of you and of your department. Figure every moment you spend talking is a moment wasted -- when your immediate goal is to learn, speaking won't achieve it. Also, since you're trying to form positive relationships, everything you say can and will be used against you later. The easiest way to avoid accidentally taking positions on important corporate issues is to avoid saying very much.
One more item to be aware of: You say your new employer is much bigger than your old one. That usually means the company politics are more complicated. Also, bigger means more opportunities for someone important to you to turn out to be a jerk of one sort or another. It's another reason to be wary of taking positions of any kind. And since you have two predecessors who failed, there's a very good chance that a jerk in high places is the reason.
There's also a good chance one of your direct reports is the reason. Not every employee begins a new reporting relationship with loyalty and a desire to help the new boss succeed.
Speaking of which, ask your staff -- both your direct reports and the rest of the employees in your organization -- to educate you, help you understand the key issues, and tell you what they think you should be doing.
If everyone in your new department learns, from day one, that you respect them and value their knowledge, ability, and good judgment, it will set a tone that will help you be the one to turn the desk o'death into a career launchpad.