# gr.group theory – Arbitrarily large finite irreducible matrix groups in odd dimension?

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 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 $$B_d$$.