It doesn't get any better than
Sins of a Solar Empire, which is a sadly overlooked game. It's exactly what you want based on what you said. The strategy is very very deep in the online multiplayer when you're playing in a team game against human opponents. However, you need to purchase the
Trinity package (or buy the game and then purchase the two expansions) so that you can play the Diplomacy expansion, which is where the 4v4 and 5v5 games are.
A stand-alone expansion called Rebellion, which will be like a Sins-2, is in the works and anticipated to be ready by the end of the year. What's nice is that it will contain all of the content of the original game and its two expansions in addition to the new Rebellion content. Also, as a new stand-alone game, it has the potential to bring hundreds of new people to the online multiplayer Sins game. Now might be a good time to get Trinity/Diplomacy so that you'll be up-to-speed when Rebellion comes out.
Sins might be best described as a 4x-RTS. Numerous Homeworld fans have played it and loved it. What's great is that it's loaded with strategic considerations, both tactical and economic, but boring micromanagement is kept to a minimum--you won't have to tell a farmer what square to farm around a city (ala Civilization or Alpha Centauri).
Here are links to two reviews that will give you a sense of the game. Note that those online multiplayer 4v4 and 5v5's only take about 1:15-1:45 to play.
http://www.youtube.com/watch?v=xY-NSNhgydY
"Imagine taking two of the greatest space games of all time, the turn-based strategy game Galactic Civilizations II and the real-time strategy game Homeworld. Imagine then merging them into one single game...It is, literally, the birth of a new genre."
http://www.youtube.com/watch?v=UyDX1CmKzC
(Do note that contrary to what the Anti-Social Fatman said in his review, unit positioning and unit micromanagement during combat is very important and can in fact win you battles, at least if your opponent isn't fully competent.)
You can find the Sins discussion forum here:
http://forums.sinsofasolarempire.com/