Using gump's reasoning I think I can prove this is impossible.

You need to connect a total of (24*23)/2=276 pairs. Each game will involve 6 pairs, and there are 4 games at any given time with 6 sessions. Therefore you have 6*6*4 = 144 "edges."

144<276, so there are not nearly enough "edges" to accomodate your pairs.

Edit: This is wrong. You would draw lines between each pair, and there would be plenty enough lines. Rather than 6-sided figures, you would have lots and lots and lots of triangles. Back to the drawing board.