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I'm having real troubles with finding the character table of the dihedral group $D_6$ of order 12: $D_6 = \langle a,b |a^6 = 1 , b^2 = 1, aba = b \rangle$. I've already found the conjugacy classes: $\{1\}, \{a, a^5\} , \{a^2,a^4 \}, \{a^3\} , \{b,ba^2,ba^4 \}$ and $\{ba,ba^3,ba^5\}$. But from there, I'm completely stuck.

Any help would be dearly appreciated.

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    $\begingroup$ Start by finding the linear characters by finding the derived subgroup. $\endgroup$ – Tobias Kildetoft Dec 31 '14 at 11:27
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Found it! For the people who need an extra hand, here's a sketch of how to do it:

First, determine the conjugacy classes. Amount conjugacy classes= amount of irreducible representations.

Second, determine these representations. The one-dimensional representations can be found via the normal subgroup properties. You'll find that there are four one-dimensional and two two-dimensional irreducible representations, you can find those by the product formula for 2 representations.

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