Generation of overdispersed poisson count data (vs gamma poisson)

[#1]

Hello,

I would like to reconcile the parametrization used in the 

Random Gamma Poisson(lambda, sigma)

with the over-dispersion parameter as estimated in generalized linear model platform.

Take for instance the attached table. Here the generating function is 

Random Gamma Poisson(exp(1+3*:X1),10)

Running glm with poisson/log + overdispersion I obtain: Overdispersion = 6.09. 

Is there any way to link lambda, sigma to the GLM over-dispersion?

thanks!
Matteo