The conjugacy classes of the quaternion group $Q_2=\langle i,i\ |\ i^4=j^4=1,\ ij(-i)=-j\rangle$ are $\{1\},\ \{-1\}, \{i,-i\}, \{j,-j\}$ and $\{ij,-ij\}.$ Proceding in a similar manner I am trying to find out the conjugacy classes of general quaternion group $$Q_n=\langle x,y\ |\ x^{2n}=1,\ x^n=y^2,\ yxy^{-1}=x^{-1}\rangle$$ of order $4n$ and the Metacyclic groups $$G=\langle x,y\ | \ x^{2^m}=1=y^{2^n}, \ yxy^{-1}=x^{-1+2^l}\rangle, \ \text{where}\ max\{2, m-n\}\leq l\leq m.$$
Please give some hints that how to proceed.
Best Answer
Let's find a conjugacy class for $x^k\in Q_n$, where $0\le k\le 2^n-1$. We have $$x^mx^kx^{-m}=x^k,\quad (x^my)x^k(x^my)^{-1}=x^{-k},$$(answer for yourself why I do not consider a conjugating by $x^my^k$ or $y^m$)
So we have $(x^k)^G=\{x^k,x^{-k}\}$. Let's do exactly the same for $y\in Q_n$.$$x^kyx^{-k}=x^{2k}y,\quad (x^ky)y(x^ky)^{-1}=x^{2k}y,$$
and it follows that $y^G=\{x^{2k}y~|~k\in\{0,\ldots,n-1\}\}$.
Can you proceed in the same manner for $x^ky\in Q_n$ and for a metacyclic group?