Row in the character table of $D_{10}$

Give the values of one row of the character table of $D_{10}$ corresponding to a character of degree $2$

I know the conjugacy classes of $D_{10}$, the dimensions of the irreducible representations and the number of irreps of each dimension.

Also the character of a representation $\pi$ is $\chi(g)=tr(g\pi)$.

Can I use this information to connstruct the table? I am really stuck on this one.

Also, does a degree 2 character mean that the vector space is of dimension $2$?

Thanks

• Yes, the degree of a character is the dimension of the underlying vector space. Moreover, do you know that $D_{10}$ can be seen directly as a subgroup of $\text{GL}_2(\Bbb R)$, since $D_{10}$ acts on the plane by a rotation and a symmetry? – Watson May 15 '16 at 19:27
• Yes, so I think I have read about some associated $2 \times 2$ matrices, with entries equal to 1 or -1, representing the rotations and reflections. So would each matrix correspond to a character? i.e. would the values in the character table be the traces of such matrices? – thinker May 15 '16 at 19:38
• Yes, the character at some $g \in D_{10}$ is the trace of the matrix associated to $g$. For instance if $g$ is the rotation of angle $2\pi/10$, your matrix is \begin{pmatrix} \cos(2\pi/10) & -\sin(2\pi/10)\\ \sin(2\pi/10) & \cos(2\pi/10) \end{pmatrix} – Watson May 15 '16 at 19:43