$D_{12}$
$${\rm LOG}\left(D_6 \otimes F_2\right)=\frac{1}{2}\pi
\left[\begin{array}{rrrrrrrrrrrr}
\bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\
\bullet & 1 & 1 & 1 & 3 & 3 & 3 & 2 & \bullet & \bullet & 2 & 2\\
\bullet & 1 & 1 & 3 & 1 & 3 & 3 & \bullet & 2 & 2 & \bullet & 2\\
\bullet & 1 & 3 & 1 & 3 & 1 & 3 & \bullet & 2 & 2 & 2 & \bullet\\
\bullet & 3 & 1 & 3 & 1 & 1 & 3 & 2 & \bullet & 2 & 2 & \bullet\\
\bullet & 3 & 3 & 1 & 1 & 1 & 3 & 2 & 2 & \bullet & \bullet & 2\\
\bullet & 3 & 3 & 1 & 1 & 3 & 1 & \bullet & \bullet & 2 & 2 & 2\\
\bullet & 3 & 1 & 1 & 3 & 3 & 1 & 2 & 2 & 2 & \bullet & \bullet\\
\bullet & 1 & 3 & 3 & 1 & 3 & 1 & 2 & 2 & \bullet & 2 & \bullet\\
\bullet & 1 & 3 & 3 & 3 & 1 & 1 & 2 & \bullet & 2 & \bullet & 2\\
\bullet & 3 & 1 & 3 & 3 & 1 & 1 & \bullet & 2 & \bullet & 2 & 2\\
\bullet & 2 & 2 & 2 & 2 & 2 & 2 & \bullet & \bullet & \bullet & \bullet & \bullet
\end{array}\right]$$
Found by P. Diţă.