Cyclic group of order four ($C_4=\mathbb{Z}/4\mathbb{Z}$)

The cyclic group generated by $r$ with $r^4=e$ and $r^2\ne e$ has elements $e,r,r^2,r^3$. It is realized by quarter-turn rotations of a plane. The cyclic group of order four and Klein four-group have the same order but different element orders: the latter has no element of order four.

For real $b,c$, this matrix defines a group representation of $C_4=⟨a:a^4=e⟩$ exactly when $bc=\pm 1$. Indeed its lower block $B$ satisfies $B^2=bc{I}_2$, so $T(a{)}^4=I_3$ exactly when $(bc{)}^2=1$. If $bc=-1$, $T(a{)}^2=\mathrm{diag}(1,-1,-1)\ne I_3$, 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.