Direct product of $D2$ and $D3$

Question

I am trying to find information on the group resulting from the direct product of the dihedral groups $D2$ (Klein four-group) and $D3$ (or, isomorphic: $S_3$ or $C_{3v}$).

What would be the resulting group? I am especially interested in the irreducible representations of the result.

Thanks!

Answer 1

If you know the representations of $S_{3}$ and $K_{4}$, then it is easy to find the irreducible representations of their direct product. It is basically the tensor product of the irreducible representations of the individual groups. So $K_{4}$ has four irreducible representations of degree 1, $S_{3}$ has two degree one irreducible representations and one degree 2 irreducible representation. So take their tensor product and obtain all the irreducible representation of the direct product. So possible degree will be 1,2 or 4. You can consult any standard representation theory book (e,g Character theory of finite group- I.M Isaacs -Chapter 4) to find a proof of this fact.