[GAP Forum] product groups

[Raghunathan,R.]

Dear GAP forum,
              Thank you for the previous responses.
I have two questions.
Question 1:
        Considering two groups 's' and 't' such that
s = ((),(3,4)) and t = ((1,3)(2,4)(5,6)), the sequential GAP commands
s:=Group((3,4));;
t:=Group((1,3)(2,4)(5,6))l;;
u:=DirectProduct(s,t);  returns 
Group([(1,2),(3,5)(4,6)(7,8)]) as the answer.
The command 
Elements(u);   returns
((),(3,5)(4,6)(7,8),(1,2),(1,2)(3,5)(4,6)(7,8)) as the answer which doesn't contain, the generators of both the groups 's' and 't' as elements. Is there an interpretation for the result?

Question 2:
        If there exist two groups 'p' and 'q' such that 
p=(generator1) and q=(generator2), are the groups 
r =(generator1,generator2) and s = {p} U {p}*{q} always equal?
      
Thank you,
Raghunathan.