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.