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I know that the Lorentz factor is given by: $$\gamma=1/\sqrt{1-v^2/c^2}$$ However $c$ is a function of the medium parameters $\epsilon_0$ and $\mu_0$. Does the Lorentz factor change for the interior of materials or is it always dependent on the speed of light in vacuum. My guess is that it is, because even the interior of a material is mostly vacuum. Have experiments to verify this been done?

eeqesri
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Lorentz factor arises in the Lorentz transformations, and the Lorentz transformations are derived using c. Therefore, the answer is:

it is always dependent on the speed of light in vacuum.

The speed of light in different media do not cause us to modify Lorentz transformations. A slowdown is just a slowdown.