Generate a sample from a multivariate Cauchy distribution

I want to generate a random sample from a multivariate Cauchy distribution, however I couldn't find a function for the multivariate Cauchy in Mathematica. I know how to define a distribution in 1D and tried to do this in a similar way

MultivariateCauchy[x_, μ_, Σ_, k_] := Gamma[(1 + k)/2]/(Gamma[1/2] π^(k/2) Sqrt[Det[Σ]] 
     (1 +(Transpose[x -μ].Inverse[Σ].(x -μ))^((1 + k)/2)))


MCdist[μ_, Σ_, k_] := ProbabilityDistribution[MultivariateCauchy[x, μ, Σ, k], {x, -∞, ∞}]

where $\mu$ is the location vector, $\Sigma$ a positive definite covariance matrix and a free scalar parameter $k$. However, this does not work with RandomVariate. Is there a way to do this?

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