To solve the equation
\[ 3 \cdot 2(x - 6)^5 = 6, \]
first, we can simplify the equation by dividing both sides by 3:
\[ 2(x - 6)^5 = 2. \]
Next, divide both sides by 2:
\[ (x - 6)^5 = 1. \]
Now, take the fifth root of both sides:
\[ x - 6 = 1. \]
Finally, add 6 to both sides to solve for \( x \):
\[ x = 1 + 6 = 7. \]
Thus, the solution to the equation is
\[ \boxed{7}. \]