Hi Npatel,
Welcome to our forum.
Amplifiers are designed for a certain output voltage (that you told us) and a certain output current. I could design a simple amplifier for you with only 1mA of output current, then find out later that it must provide 1000 times more. A completely different amplifier would be required but both amplify the voltage the same amount.
Since you also don't say how much input current the amplifier is allowed to draw from the source, I could design a simple amplifier for you that might overload the source, then find out later that it must have a much higher input resistance.
Also, the amplifier won't be simple and will need a compressor to reduce its gain with higher input voltages. You want to amplify 1.2V to 6V, which is a gain of 5 times. But you want to amplify 4V to only 9V, which is a gain of only 2.25 times. Is this correct?
Including the compressor, it would take a few transistors to make the amplifier. A single opamp would do it easily.
Please provide us with all the details for the amplifier: output current, input current and whether a compressor is required.