Given group $T$ of order $8$, and $t \in T$ such that $ord(t) = 4$.
Let $P = \{1,t,t^2,t^3 \}$ and let $x \in TâP$.
List possibilities for $x^2$ labelling as $(a_1,a_2, \ldots ,a_n)$.
List possibilities for $xyx^{-1}$ labelling as $(b_1,b_2, \ldots ,b_n)$.
By considering each $(a_i,b_j)$, show that $T$ is isomorphic to $C_8$, $C_4 \times C_2$, $Q_8$ or $D_8$.
I know that $P$ is normal in $T$, also $x^2 \in T$ and I believe that $xyx^{-1} \in T$.
Not sure what to do next.
