Hmmm. BTW is there a more down to earth approach?

That's great! so how can we construct the period matrices of simple complex tori for any dimension $g \geq 2?$

Thanks for your response. That does not really answer my question. I think, I should have been more precise. By algorithm, I mean how to construct the period matrices associated with simple complex tori/abelian varieties. Moreover, the method you described was explained in the above given reference.

@Steve D: That's a good point! I mean, for each sides of the semi-product, and wreath product (which I'm not very familiar and comfortable with the latter one)

@Yemon Choi: I tried for the semi-direct product of finite groups, but nothing came out!

If you ask Behrend for giving you an example, it will end up in sort of new prime numbers, which will be called Behrend's prime numbers!

@Howard M Thompson: Thanks for your response.