I have been struggling computing the Fisher's information of the Wishart distribution. I'll write what I have gone through. Let's $\Omega$ denote a $p\times p$ Wishart random variate denoted by $\mathcal{W}(k,V)$ where $k$ is the degrees of freedom and $V$ a positive definite scale matrix. If we write $\mathcal{W}(\Omega\,|\,k,V)$ for the density function, $$ \begin{align} \nabla_{\operatorname{vech}(V)}\log\mathcal{W}(\Omega\,|\,k,V) &= \dfrac{1}{2}D_{p}'\left(V^{-1}\otimes V^{-1}\right)D_{p}\operatorname{vech}(\Omega)-\dfrac{k}{2}D_{p}\operatorname{vech}\left(V^{-1}\right)\\ \nabla_{k}\log\mathcal{W}(\Omega\,|\,k,V) &= \dfrac{1}{2}\log|\Omega|-\dfrac{1}{2}\log|V|-\dfrac{p}{2}\log 2-\dfrac{1}{2}\sum_{i=1}^{p}\psi\left(\dfrac{k+1-i}{2}\right) \end{align} $$ where $D_{p}$ is the unique duplication matrix such that $D_{p}\operatorname{vech}(A)=\operatorname{vec}(A)$, $\otimes$ Kronecker product, and $\psi$ digamma function.
What is the Fisher information matrix of the Wishart distribution?