= Solution
Define the spinor generators
$$
S^{\mu\nu}=\frac14[\gamma^\mu,\gamma^\nu].
$$
Repeated use of the <Clifford algebra> gives
$$
\begin{aligned}
[S^{\mu\nu},\gamma^\rho]
&=\frac14(\gamma^\mu\gamma^\nu\gamma^\rho
-\gamma^\nu\gamma^\mu\gamma^\rho
-\gamma^\rho\gamma^\mu\gamma^\nu
+\gamma^\rho\gamma^\nu\gamma^\mu)\\
&=\gamma^\mu\eta^{\nu\rho}-\gamma^\nu\eta^{\rho\mu}.
\end{aligned}
$$
The <Spinor representation of the Lorentz group> is
$$
S[\Lambda]=\exp\left(\frac12\Omega_{\rho\sigma}S^{\rho\sigma}\right).
$$
The adjoint relation for the Dirac matrices implies
$$
(S^{\rho\sigma})^\dagger=-\gamma^0S^{\rho\sigma}\gamma^0.
$$
Exponentiating yields
$$
\boxed{S[\Lambda]^\dagger=\gamma^0S[\Lambda]^{-1}\gamma^0}.
$$
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