The previous answer by Ottem shows that there is some way to write the generating function to correspond to a chain of syzygies. However, it is not true that every way works. For instance, let $G$ be the group of order eight generated by the two $3\times 3$ diagonal matrices with diagonals $(-1,-1,1)$ and $(1,1,i)$ (where $i^2=-1$). A "correct" way to write the generating function is $(1+t^2)/(1-t^2)^2(1-t^4)$. An "incorrect" way is $1/(1-t^2)^3$. Note that the series given in aglearner's question can also be written incorrectly by cancelling $1-x^6$ from the numerator and denominator.
The previous answer shows that there is some way to write the generating function to correspond to a chain of syzygies. However, it is not true that every way works. For instance, let $G$ be the group of order eight generated by the two $3\times 3$ diagonal matrices with diagonals $(-1,-1,1)$ and $(1,1,i)$ (where $i^2=-1$). A "correct" way to write the generating function is $(1+t^2)/(1-t^2)^2(1-t^4)$. An "incorrect" way is $1/(1-t^2)^3$.