This is a cross post from MSE, as it seems the partial answer I got then was deleted, so I ask it again here.

Let $\prod_{i\in I}p_{i}^{a_{i}}$ be the prime factorization of a positive integer $n$ and let's consider the $\Omega(n)$-parallelotope built with, for all $i\in I$, $a_{i}$ pairwise orthogonal vectors of norm $p_{i}$, so that there are $\omega(n)$ different lengths for the sides of this parallelotope. Let's consider the isometry group of this parallelotope, as a part of the affine space $\mathbb{R}^{\Omega(n)}$, denoted by $G(n)$.

Can this group provide information of arithmetical interest about $n$? I'm especially interested in estimating the size of $ \pi_{G}(x) : =\{n\leq x, G(n)\cong G\} $.

This is a cross post from MSE, as it seems the partial answer I got then was deleted, so I ask it again here.

Let $\prod_{i\in I}p_{i}^{a_{i}}$ be the prime factorization of a positive integer $n$ and let's consider the $\Omega(n)$-parallelotope built with, for all $i\in I$, $a_{i}$ pairwise orthogonal vectors of norm $p_{i}$, so that there are $\omega(n)$ different lengths for the sides of this parallelotope. Let's consider the isometry group of this parallelotope, as a part of the affine space $\mathbb{R}^{\Omega(n)}$, denoted by $G(n)$. Of course, any permutation of $ I $ gives rise to the same group, and as such should be seen as an automorphism thereof.

Can this group provide information of arithmetical interest about $n$? I'm especially interested in estimating the size of $ \pi_{G}(x) : =\{n\leq x, G(n)\cong G\} $.