I'm studying group theory and now I'm analyzing non-abelian groups of order 12. I see that the dihedral group $D_6$ can be expressed by
$$D_6=\langle a,b : a^6=1, b^2=1, aba=b\rangle =\langle 1,a,a^2,a^3,a^4,a^5, b, ba, ba^2, ba^3, ba^4, ba^5\rangle,$$
and the alternating group $A_4$ by
$$A_4=\langle 1, (123),(132),(124),(142),(134),(143),(234),(243),(12)(34),(13)(24), (14)(23)\rangle.$$
I want to know what are the elements of the dicyclic group $\text{Dic}_3$ defined as
$$\text{Dic}_3=\langle a,b : a^6=1, b^2=a^3, bab^{-1}a=1\rangle.$$
Any help would be appreciate.
[Math] Elements of the dicyclic group of order 12
group-theory
Best Answer
Just to follow on from my comment:
Notice that the subgroup $\langle a\rangle$ is normal in ${\rm Dic}_3$, so the elements can be labelled in the same way as in $D_6$, though obviously the binary operation will be different.
In fact it is relatively easy to check that ${\rm Dic}_3$ is generated by $a^2$ and $b$ and that $\langle a^2$ is a normal subgroup of order $3$, so ${\rm Dic}_3\cong C_3\rtimes C_4=\langle x,y|x^3=y^4=1,yxy^{-1}=x^{-1}\rangle$. This is similar to the description $D_{6}\cong C_6\rtimes C_2$ that makes dihedral groups perhaps a little more tangible.
I noticed this by chance, but could have found it by looking for the Sylow $3$-subgroups. This is often a good place to start when trying to understand a new finite group.
Alternatively, for very tangible elements, one may identify ${\rm Dic}_3$ with the subgroup $\langle (1,2,3),(2,3)(4,5,6,7)\rangle$ of $S_7$.