TITLE: Conjugacy classes of the nonabelian group of order 21
QUESTION [8 upvotes]: How would you go about finding the conjugacy classes of the nonabelian group of order 21, $G:=\left\langle x,y | x^7=e=y^3, y^{-1}xy=x^2\right\rangle$?

REPLY [2 votes]: Here order of the group is $21= 3 \times 7$ . It is easy to see $3$ divides $7-1=6$. Thus, we have (upto isomorphism)two groups of order $21$. One of them is cyclic $\mathbb{Z}_{21}$ and other is non-abelian. This non-abelian is generated by two elements say $a$ and $b$ such that $|a|=3$ and $|b|=7$ and $ba=ab^r$ where $r$ is not congruent to $1$ modulo $7$ and $r^3\equiv 1 (mod 7)$. Defining conjugacy it is easy to see that there are five congugate classses and center has only the identity element. Moreover, class equation is $1+3+3+7+7\ldots$ :)