need examples of different groups

Question

I need examples of different groups having different properties, such as:

1. class 2 or 3
   
2. cyclic commutator
   
3. cyclic center
   
4. $Z(G)\le \Phi(G)$
   
5. redei group
   
6. $G=\langle aG',bG' \rangle $
   

and ...
Are there books or websites that have this kind of example?
because most books I read online are just filled with theorems and lemmas.

Answer 1

As an online resource, you could have a look at this website:

http://ericmoorhouse.org/pub/bol/.

This is about the weaker (not necessarily associative) structure of some finite loops but, among these, you can search the isotopy classes of groups. For example

http://ericmoorhouse.org/pub/bol/htmlfiles8/8_5_2_0.html

or

http://ericmoorhouse.org/pub/bol/htmlfiles18/18_1_18_1.html.

Below each Cayley table you find several things like, e.g., the elements of the centrum (set of all elements of the group $G$ which commute with every element of $G$), the order of each element of $G$ and some other facts interesting for groups.