Solution:

To find the dimensions of the gravitational constant \( G \), use Newton's law of gravitation:

\[
F = \frac{G M_1 M_2}{r^2}
\]

Where:
- \( F \) is force (with dimensions \( [M L T^{-2}] \)),
- \( M_1 \) and \( M_2 \) are masses (with dimensions \( [M] \)),
- \( r \) is distance (with dimensions \( [L] \)).

Rearranging for \( G \):

\[
G = \frac{F r^2}{M_1 M_2}
\]

Substitute the dimensions:

\[
G = \frac{[M L T^{-2}] [L^2]}{[M][M]}
\]

Simplify:

\[
G = [M^{-1} L^3 T^{-2}]
\]

Thus, the dimensions of \( G \) are \( [M^{-1} L^3 T^{-2}] \).