To find the sum of the first three terms of a geometric progression (G.P.) where the first term (\(a\)) is 6 and the fourth term is 162, follow these steps:

1. Identify the given values and the common ratio (\(r\)):

- The first term \(a = 6\).
- The fourth term is given by \(a \cdot r^3 = 162\).

2. Set up the equation for the fourth term:

\[
6 \cdot r^3 = 162
\]

Solve for \(r^3\):

\[
r^3 = \frac{162}{6} = 27
\]

Find \(r\):

\[
r = \sqrt[3]{27} = 3
\]

3. Calculate the first three terms of the G.P.:

- The first term is \(a = 6\).
- The second term is \(a \cdot r = 6 \cdot 3 = 18\).
- The third term is \(a \cdot r^2 = 6 \cdot 3^2 = 6 \cdot 9 = 54\).

4. Find the sum of the first three terms:

\[
\text{Sum} = 6 + 18 + 54 = 78
\]

Thus, the sum of the first three terms of the geometric progression is 78, which corresponds to option D.