I know characters of two 2-dimentional irreps (U and V) of a group with 6 conjugate classes. The characters are: $\begin{pmatrix} 2&-1&-1&2&0&0\end{pmatrix}$ and $\begin{pmatrix} 2&1&-1&-2&0&0\end{pmatrix}$. Sizes of conjugate classes are 1,2,2,1,3,3 respectively.
I want to deduce decomposition of $U\otimes V$ into sum of irreps from this information. Is it possible? Do I need Clebsch-Gordan series for this?
The character of the representation $U\otimes V$ is $(4\ -1\ 1\ -4\ 0\ 0)$. We can calculate the following inner products $$ \langle \chi_{U\otimes V},\chi_{U\otimes V}\rangle=\frac1{12}(16+2+2+16)=3. $$ This implies that $U\otimes V$ is a sum of three pairwise non-isomorphic irreducible representations. One of the components is thus 2-dimensional and the other two are 1-dimensional.
$$ \langle \chi_{U\otimes V},\chi_{U}\rangle=\frac1{12}(8+2-2-8)=0, $$ and $$ \langle \chi_{U\otimes V},\chi_{V}\rangle=\frac1{12}(8-2-2+8)=1, $$ so we can deduce that the 2-dimensional component is isomorphic to $V$. No further information can be deduced from the given data. But there are relatively few isomorphism types of groups of order 12, so it is not a stretch to think that we might be able to positively identify it. Leaving that to somebody else :-)