# Thread: irreducible representations of the quaternion group

1. ## irreducible representations of the quaternion group

Find all the irreducible representations of the quaternion group and write its character table.

2. Originally Posted by akc2010
Find all the irreducible representations of the quaternion group and write its character table.
Call the quaternion group Q. I will say irrep for an irreducible representation and rep for a representation.
Character table:
To get its character table first show that Q has 5 conjugacy classes.
Then find all its 1 dimensional reps as given below..
First prove that $[Q,Q] = \{\pm 1\}$. We know that all the one dimensional reps of Q can be obtained from the one dimensional reps of Q/[Q,Q] (which is isomorphic to Klein 4 group). I assume you know the one dimensional reps of Klein 4 group. (hint: Just try all feasible combination of 1 and -1 in the table).
Then use sum of (character of identity)^2 = #Q = 8 to figure out the only non 1 dimensional irrep's dimension is 2.
Now use orthogonality relations to fill the table. You will observe non 1 dimensional rep's character has a peculiar nature which will help you figure out the irrep