I am looking for finite groups that have a 2D (complex matrix) faithful irreducible representation and a 3D faithful irreducible representation. Up to order 1023 GAP found none. Other combinations of dimensions seem to occur. Are groups with 2D and 3D faithful irreps non-existing?
Can a finite group have 2D and 3D faithful irreducible representations
group-theoryrepresentation-theory
Best Answer
Here is a proof that no finite group has faithful irreducible representations of degrees $2$ and $3$. It might be easier to do this using the classification, of finite subgroups of ${\rm G}(2,{\mathbb C})$, but I am more familiar with the subgroups of ${\rm GL}(2,q)$ for finite $q$, so I will use that approach.
Let $G$ be a finite irreducible subgroup of ${\rm GL}(2,{\mathbb C})$. Let $p$ be a prime not dividing $|G|$. Then, by standard results in representation theory, the associated complex representation can be written over the finite field of order $q$ for some power $q$ of $p$.
The irreducible subgroups of ${\rm GL}(2,q)$ are either imprimitive or semilinear, or they have normal subgroups isomorphic to ${\rm SL}(2,3)$ or ${\rm SL}(2,5)$.
The imprimitive and semilinear groups have a normal abelian subgroup of index $2$, and since all complex irreducible representations of abelian groups have degree $1$, the irreducible representations of $G$ have degree at most $2$.
On the other hand, ${\rm SL}(2,3)$ or ${\rm SL}(2,5)$ do not have faithful irreducible representations of degree $3$.