How to determine the isomorphism types of given groups with generators and relations

Question

I was classifying the all groups of order 30 and I got the following groups
$\langle a,b \mid b^{-1}ab=a^4, a^{15}=b^{2}=1\rangle$ and $\langle a,b \mid b^{-1}ab=a^{11}, a^{15}=b^{2}=1\rangle$.

How can I show that these groups are isomorphic to $\mathbb{Z}/3\mathbb{Z}\times D_{10}$ and $\mathbb{Z}/5\mathbb{Z}\times D_{6}$? Thanks!.

Answer 1

For the first one, let $H = C_3 \times D_{10} = \langle x \rangle \times \langle y,z \rangle$ with $|x|=3$, $|y|=5$, $|z|=|yz|=2$.

Check that $(xy)^{15} = z^2=1$, and $z^{-1}(xy)z = (xy)^4$. Deduce that there is a homomorphism $\phi:G \to H$ with $\phi(a)=xy$, $\phi(b)=z$.

Show that $H = \langle xy,z \rangle$ and deduce $\phi$ is surjective.

Use the relations of $G$ to show that $|G| \le 30$ and deduce that $\phi$ is injective, and hence an isomorphism.

The second one is similar.