Identify the presentation of groups

Question

I'm trying to classify all groups of order $20$. I get two group presentation and want to identify isomorphism type. I think one of these isomorphic to $D_{10}$ and other isomorphic to $Dic_5$. Can anyone tell me to find exactly which one isomorphic to $D_{10}$ and other isomorphic to $F_5$? I might get wrong presentations.

These are my two presentation,

$$G_1 =\langle r,a,b\mid r^5=1,a^2=b^2=1,r^{3}br^{-3}=b,r^{3}ar^{-3}=a\rangle$$

and

$$G_2 =\langle r,s\mid r^5=1,s^4=1,r^3sr^{-3}=s\rangle $$

Thank you.

Answer 1

Here is a small collection of posts about the classification of groups of order $20$. It is too long for a comment.

1. Find four groups of order 20 not isomorphic to each other.

2. Four groups of order 20 that are not isomorphic

3. Classifying groups of order $20$

4. Classification of groups of order $20$

5. Must a group of order $20$ have an element of order $10$?

6. Prove G is a nonabelian group of order 20

7. How can I show that $G$ is non abelian of order 20?

The last one has the presentation for $F_5$, or sometimes denotes Frobenius $F_{20}$, namely $$ G=\langle x,y|x^4=y^5=1,x^{-1}yx=y^2\rangle $$