If a, b, and c form a geometric progression then b = ar and c = ar^2 (for some ratio r):
abc &= 6^6 \\
a(ar)(ar^2) &= 6^6 \\
a^3 r^3 &= 6^6 \\
ar &= 36 \\
b &= 36
\(\displaystyle b - a\) has to be a perfect cube, or \(\displaystyle 36 - a\) has to be a perfect cube. There are a few possibilities for a (35, 28, 9), but the only one that will work in our case is if a = 9.
If a = 9 and b = 36, then r = 4, which makes c = 144. So a + b + c = 189.