The Life of a Mathematician

Okay, let's give this a try. For my representation theory class, I'm doing a presentation on a family of groups. For a positive integer n, the group T4n is defined by:

<a,b a^(2n) = 1, a^n = b^2, aba = b&gt

After playing around with this for a while, I found a convenient way to list the 4n elements for a given n:

1, a, a^2, a^3, ... , a^(2n-1), b, ab, (a^2)b, ... , (a^(2n-1))b

The next step was to find the conjugacy classes. It turns out there are n + 3 of them.

Two have just one element:

{1}, {a^n}

n-1 have two elements:

{a, a^(2n-1)}, {a^2, a^(2n-2)}, ... , {a^(n-1), a^(n+1)}

Two have n elements:

{(a^t)b t is even}, {(a^t)b t is odd}

From there I've managed to find all the one-dimensional representations (there are four of them for any given group), and I know that the remaining n-1 irreducible representations are all two-dimensional. I believe that I know what those representations are, but I haven't proved that they are irreducible yet.