I consider a finite irreducible matrix group $Gammasubseteqmathrm{GL}(Bbb R^d)$. 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 choose the symmetry group of the $n$-th cartesian power of a regular $k$-gon:

$$P=overbrace{C_ktimes cdotstimes C_k}^{text{n times}}.$$

This group is irreducible and gets arbitrarily large with $ktoinfty$.

Question:What about odd dimensions? Can there be arbitrarily large finiteirreduciblematrix 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 $B_d$.