To find the perimeter of the dilated triangle, we first need to calculate the perimeter of the original triangle with side lengths of 6, 6, and 5.
The perimeter \( P \) of the original triangle is given by the sum of its side lengths:
\[ P = 6 + 6 + 5 = 17 \]
Now, when the triangle is dilated with a scale factor of 4, each side length is multiplied by 4. Therefore, the new side lengths of the dilated triangle will be:
\[ 6 \times 4 = 24, \] \[ 6 \times 4 = 24, \] \[ 5 \times 4 = 20. \]
Now we can calculate the perimeter of the dilated triangle:
\[ P_{\text{dilated}} = 24 + 24 + 20. \]
Calculating this gives:
\[ P_{\text{dilated}} = 24 + 24 + 20 = 68. \]
Thus, the perimeter of the dilated triangle is \( \boxed{68} \).