= Solution
Use the axes actually printed: the two hydrogens lie in the $xz$ plane, $z$ is the principal twofold axis, and $y$ is perpendicular to that plane. For a Cartesian-displacement <group representation>, only atoms left at their own sites contribute to the <trace>; their vector components must also be transformed. In the order $(E,C_2(z),\sigma(xz),\sigma(yz))$, identity fixes six components, the twofold rotation exchanges the hydrogens, the $xz$ reflection fixes both hydrogens with vector <matrix> $\operatorname{diag}(1,-1,1)$, and the $yz$ reflection exchanges them. Therefore
$$
\boxed{\chi_\Gamma=(6,0,2,0).}
$$
The nonzero mirror <character of a representation> is $2(1-1+1)=2$, not six.
Back to article page
