Not including the identity transformation, the eleven transformations that preserve the regular hexagon shown are counterclockwise rotations by $60^\circ,$ $120^\circ,$ $180^\circ,$ $240^\circ$ and $300^\circ$ and reflections across the six dashed lines shown. Kristina randomly picks six transformations $T_1,$ $T_2,$ $T_3,$ $T_4$, $T_5$ and $T_6,$ with replacement, from this set of eleven. She performs these six transformations on the hexagon, in succession. The probability that the point $P$ is transformed to each of the hexagon's six vertices exactly once during this process is $\dfrac{k}{11^6}.$ What is the value of $k\,?$