I would like calculate the number of the different complex numbers that we need in order to describe the general interaction for the fourth and fifth order nonlinearity, how I can do that?

How many different complex numbers are we need in order to describe the general interaction for the fourth and fifth order nonlinearity if the Intrinsic permutation symmetry is applied?

How many different complex numbers are we need in order to describe the general interaction for the fourth and fifth order nonlinearity if the Kleinmain's symmetry is applied?

How many different complex numbers are we need in order to describe the general interaction for the fourth and fifth order nonlinearity if the crystal material is isotropic?

Thank you in advance!

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