Represent the table as a clock with people seated at 2, 4, 6, 8, 10, 12 hours.

Let the people be represented as letters A-F. Let also A, B and C identify those people who do not wish to sit together and put them in table positions 2, 6 and 10, so that D, E and F sit in positions 4, 8 and 12. A, B and C can shift positions in 6 different ways, and so can D, E and F. For every position of A, B and C, there are 6 different positions for D, E and F, therefore there are 36 different arrangements. But each unique arrangement of people can start at a different seat: 2, 4, 6, 8, 10, 12, making the total number of arrangements=6×36=216.