I am cross-posting this question from my MSE post here, in case someone here can answer it.
For a finite group $G$, the rational group ring $\mathbb{Q}[G]$ has a Wedderburn decomposition: $$ \mathbb{Q}[G] \cong \prod_{i=1}^k M_{n_i}(D_i), $$ where $D_i$ is a division algebra whose center is a number field, and $M_{n_i}(D_i)$ is the ring of $n_i \times n_i$ matrices over $D_i$. I will refer to the factors of this product as the simple factors of $\mathbb{Q}[G]$. Each simple factor corresponds to an irreducible $\mathbb{Q}$-representation of $G$.
My question is: given $n$ and a primitive $d$th root of unity $\zeta_d$, for which finite groups $G$ does $M_n(\mathbb{Q}(\zeta_d))$ appear as a simple factor of $\mathbb{Q}[G]$ corresponding to a faithful representation of $G$?
It is well known that if $G$ is cyclic of order $d$, then the unique faithful irreducible $\mathbb{Q}$-representation of $G$ has $\mathbb{Q}(\zeta_d)$ as its corresponding simple factor. Moreover, if $G$ is a Heisenberg group over $\mathbb{Z}/p\mathbb{Z}$, then it has a unique faithful irreducible representation over $\mathbb{Q}$ and the corresponding simple factor is $M_p(\mathbb{Q}(\zeta_p))$; Kenta S sketched a proof of this in the MSE thread.
What other examples are there?