**Example 3**. Irreducible Representations of : In part 5) in here we showed that the dihedral group of order has five non-equivalent irreducible representations. Four of them have degree one and one of them has degree two. So let Every element of can be written uniquely as where and

1) Representations of degree one. These have been completely described in Example 2 in this post. In our case which is an even number. So there are four representations of degree one, say They are defined on by

where

2) Representation of degree two. Here we described representations of degree two for the dihedral group We proved there that all except those corresponding to are irreducible. In our case So there are two irreducible representations and we’ll pick the one corresponding to Let’s call it So if then is defined on by

for all and I’ll leave it to the reader to find an explicit formula for

**Example 4**. Irreducible Representations of : In part 4) in here we showed that the quaternion group of order has five non-equivalent irreducible representations. Four of them have degree one and one of them has degree two.

1) Representations of degree one. These were fully described in Example 1 in this post. We showed that if is a degree one representation of then

Therefore and similarly Finally, since we have and

2) The representation of degree two. So we need matrices with entries in which satisfy the same relations as do. Let and Let Now if we define and Since satisfy the same relations as do, will give us a group homomorphism from to and hence it is a representation of degree two. So, for example etc. It is easy to see that have no common eigenvector and thus is irreducible by the theorem we proved in here.