There are six ways to arrange a group of three people such that they are still sitting together:

abc, bac, cab, acb, bca, cba

Then you need to multiply that 6 by the ways you can seat the other three people around that group. You can have them all on the left of the first group of three (6 different permutations of people to achieve that), all on the right side (another six ways) two on the left and one on the right (six more ways) and one on the left and two on the right (a final six ways). So 6 x (6+6+6+6) is 6 x 24 = 144