The product $d\times d^*$ cannot be "so" large, as a consequence of the so-called Transference theorems Particularly, Thm. 2.1. of the paper shows that $d\times d^* \leq n^2$ (hence in your example, $n$ should be at least $8$). So the best you can do in terms of the product is $\Omega(n^2)$. The aforemetioned Conway-Thompson Theorem (see [1], p. 46) shows that indeed it *is* possible to achieve $\Omega(n^2)$.

In the case of $d, d^* \geq 8$, we have the lower bound $n \geq 8$ and the achievable upper bound $72$ (due to the unimodular table). Nevertheless, there is still a huge gap between both bounds. I am not aware of any result that shows that smaller $n$ are possible.

[1]. J. Milnor and D. Husemoller, ``Symmetric Bilinear Forms,'' Springer-Verlag, New York,
1973