Free Product (Coproduct, or Categorical Sum)
Each multiplicand is represented by a 'word' consisting of a sequence of generators. We apply the multiplication by putting the words one after the other. This is the most general type of product that contains the two multiplicands as subgroups.
We can 'reduce' the result by:
 removing the identity element
 replacing and sequence that is within one of the multiplicands
but unless we apply any specific reductions the result will always be infinite for non trivial multiplicands.
Example C_{2}×C_{3}
In order to try to understand this product of two groups lets try multiplying two very simple groups together, the simplest groups I can think of are C_{2} and C_{3}
C_{2}
generator  cayley graph  table  

<m  m²> 

C_{3}
generator  cayley graph  table  

<r  r³> 

Free Product C_{3} × C_{2}
So what would be the elements of this group? Lets assume they are the same as for the direct product, that is: 1,r,r²,m,rm and r²m. Using these as words would give the table:
1  r  r²  m  rm  r²m 
r  rr  rr²  rm  rrm  rr²m 
r²  r²r  r²r²  r²m  r²rm  r²r²m 
m  mr  mr²  mm  mrm  mr²m 
rm  rmr  rmr²  rmm  rmrm  rmr²m 
r²m  r²mr  r²mr²  r²mm  r²mrm  r²mr²m 
We can then apply any constrains for the individual groups, that is: m²=1 and r³=1 which gives:
1  r  r²  m  rm  r²m 
r  r²  1  rm  r²m  m 
r²  1  r  r²m  m  rm 
m  mr  mr²  1  mrm  mr²m 
rm  rmr  rmr²  r  rmrm  rmr²m 
r²m  r²mr  r²mr²  r²  r²mrm  r²mr²m 
But we still get elements that are not members of the group (the group is not closed). If we make these elements of the group, then we have to make them rows and columns, in which case more elements get added to the body of the table. So the group is infinite. The elements of the group are alternating r's and m's such as:
r,m,r,m,r,m,r,m…
or
m,r,m,r,m,r,m,r…
where r may be replaced by r² at any place.
generator  cayley graph  table 

<m,r  m²,r³> 