I have to show that the two-dimensional irreducible representation of the quaternion group is isomorphic to an induced representation

, where

is a subgroup of

of index two and

is a one-dimensional character.

For starters, I don't even know the two-dimensional irreducible representation of the quaternion group! And I'm not really sure how to progress past that, even, since I'm not comfortable with the induced representation over

and not a vector space...any help is much appreciated!

About the only thing I realize here is that the subgroup H is normal in Q...but I'm not even sure if or how that helps me.