Required: surface area

Relevant wiki: Surface Area - Problem Solving

From my diagram, in order to compute for the slant height $L$ , we need to know $x$ . By pythagorean theorem, $x=\sqrt{6^2-3^2}=3\sqrt{3}$ . So, $L=\sqrt{12^2+(3\sqrt{3})^2}=3\sqrt{19}$

The surface area is equal to the lateral area (sum of the areas of the triangles) plus the area of the hexagon. The lateral area is $\dfrac{1}{2}PL$ where $P$ is the perimeter of the base and $L$ is the slant height. The area of the hexagon is $\dfrac{3}{2}\sqrt{3}x^2$ where $x$ is the edge length of the hexagon. Thus,

$A=\dfrac{1}{2}(6)(6)(3\sqrt{19})+\dfrac{3}{2}\sqrt{3}(6^2)=54\sqrt{19}+54\sqrt{3}=54(\sqrt{19}+\sqrt{3})$

So the desired answer is $a+b+c=54+19+3=\boxed{76}$