The left-hand side of $G^{\circ}\times H \rightarrow G$ should be a semi-direct product, right?

Thanks for this answer.

@Fernando: I am extremely sorry for that, but now the question has been edited by Stefan Kohl and it has been asked in an appropriate language.

I was interested in the determinant (mod $p$) and the trace (mod $p$) of a matrix defined as follows :- For each prime $p$, define a $(p-1)*(p-1)$ integer matrix $M^{p}$ as $M^{p}_{ij} \equiv i^{j}$ (mod $p$), $0<i<p, 0<j<p$. I was surprised to see that the trace was 0 (mod $p$) only for 17 and 19 in the first 3000 primes and thus I thought of posting this question. (I know that 3000 is very small number and one should not make any blind conjecture based on only 3000 evidences).