jmc

Utilizing the properties of geometric sequences, we obtain: $$a^2=6b\text{ and }{a}^2=\frac{54}{b}.$$Thus, $6b=\frac{54}{b}$, and $b=3.$

Plugging that into the first equation, we have $a^2=18$, meaning $a=\boxed{3\sqrt{2}}$