the same amount of channels as with any other! The size has nothing to do with the number of "channels" are received or transmitted. The size of the reflector is directly related to the amount of gain needed to do the job.
The big boy is one I retrofitted with a new Monopulse Servo Control System and new Low Noise Amplifier system for the Intelsat Gateway Earth Station in Fuschstadt Germany a few years back. It's mission is Tracking Telemetry and Control of a satellite. The spacecraft flight engineers use it to keep the satellite itself in the correct geosynchronous orbit.
Occasionally, it's also used for In Orbit Transfer (IOT), of a new satellite to its final orbital slot after launch from a rocket.
It's 34 meters in diameter and weighs about 600 tons.
With the improvements in satellite technology and gound station electronics in the last 20 years, antennas this large are not actually required anymore. But its less expensive to retrofit them and get another 10 years service, than it is to tear them down and replace it with a smaller (16 meter) 'new and improved' one.
The other "smaller" antennas to the left are not puppies. They are 13 meters diameter and go in at about 55 tons. The 13 meter antennas are comms antennas for video, voice and data traffic to the Mid East, parts of Asia and Africa. They both receive and uplink the traffic.
Wow! How many channels do you get with that set-up.