Conjugacy classes of the group $G=\langle a,b\mid a^6=1,b^2=a^3,bab^{-1}=a^{-1} \rangle$ Consider the group $G=\langle a,b\mid a^6=1,b^2=a^3,bab^{-1}=a^{-1} \rangle$. This group does not appear to be easy to work with! Does anyone know what this group is called?
I am trying to find its conjugacy classes.
I think I am right in saying that $\{e\}$ is always a conjugacy class. So I now consider $a^{j}a^{i}a^{-j}=a^{i} \ \ \ \forall i$, and, $b^ja^ib^{-j}$ which I believe needs to be considered by cases. So for $i>j$ I have $b^ja^ib^{-j}=a^{i(-1)^j}$ and for $i \leq j$ I have that $b^{j-i}a^{i(-1)^j}b^{i-j}$. This is not very tractable.
You should be able to swap a and b's in the answer to the above to obtain other conjugacy classes of $b^j$. 
What I am doing wrong? I have a bad feeling about my method.
 A: Note that $\langle a \rangle$ is a normal subgroup of $G$, and $G/\langle a\rangle$ is of order $2$. This means $G$ must have order $|G|=|\langle a\rangle||G:\langle a\rangle |=6\times 2$. It is not $A_4$ since it has an element of order $6$, you can see it is not $D_{12}$ either since $D_{12}$ has no element of order $4$. It follows it is the unique nonabelian group $T$ of order $12$ which is not the former (there are three nonabelian groups of order $12$). 
This group is best described as a semidirect product $C_3\rtimes C_4$ where $C_4$ acts on $C_3$ by inversion. Hence your group has a presentation $\langle j,k\mid j^3,k^4,kjk^{-1}=j^{-1}\rangle$. 
Note that from $kjk^{-1}=j^{-1}$ we have $k^2jk^{-2}=j$, so that $k^2$ commutes with $j$. It follows $k^2$ commutes with $j^2$. Now $k^2$ has order $2$ and $j^2$ and order $3$, so $a=j^2k^2$ has order $6$. Moreover, $a^3= k^2$, and $k$ has order $4$. You can check then that $\eta:T\to C_3\rtimes C_4$ with $a\to j^2k^2$ and $b\to k$ is an isomorphism. 
A: This is the dicyclic group of order 12:
http://en.wikipedia.org/wiki/Dicyclic_group
http://groupprops.subwiki.org/wiki/Dicyclic_group
As for the conjugacy classes, as others said, it's a good exercise. (Hint: there are six.)
