= Faithfulness criterion for a block representation of the cyclic group of order four
{title2=$T(a)=\operatorname{diag}\!\left(1,\begin{pmatrix}0&b\\c&0\end{pmatrix}\right)$}
For real $b,c$, this matrix defines a <group representation> of $C_4=\langle a:a^4=e\rangle$ exactly when $bc=\pm1$. Indeed its lower block $B$ satisfies $B^2=bcI_2$, so $T(a)^4=I_3$ exactly when $(bc)^2=1$. If $bc=-1$, $T(a)^2=\operatorname{diag}(1,-1,-1)\ne I_3$, so its order is four and the representation is <faithful>. If $bc=1$, its order is exactly two and the <kernel of a group homomorphism> is $\{e,a^2\}$, giving an unfaithful representation. Other values do not give representations at all, including $bc=0$, when the proposed matrix is singular.
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