I consider a finite irreducible matrix group $\Gamma\subseteq\mathrm{GL}(\Bbb R^d)$. I am interested in the maximal size of $\Gamma$ depending on $d$. But this question makes only sense if there is an upper limit.
In even dimension there is no such limit. This is easiest seen in dimension $d=2$, where we have the cyclic groups or dihedral groups of arbitrarily large size. More generally, in dimension $d=2n$ we can consier the $n$-th cartesian power of a regular $k$-gon $P_k$:
$$\overbrace{P_k\times \cdots\times P_k}^{\text{n times}}.$$
Its symmetry group is irreducible and gets arbitrarily large with $k\to\infty$.
Question: What about odd dimensions? Can there be arbitrarily large finite irreducible matrix groups in dimension $d=2n+1$?
For example, in dimension $d=3$ we have the arbitrarily large symmetry groups of prisms and antiprisms, which are reducible. The largest irreducible group is probably the symmetry group of the icosahedron.
I have the feeling that in sufficiently large odd dimensions, the largest such group is the reflection group $B_d$.