Before continuing, prove or disprove: If is a group, the direct power is never generated by fewer than elements.
Certainly this is the case if is abelian, as in this case has a quotient of the form , i.e., an -dimensional vector space over . This is also the case if . Examples of small size are therefore a little hard to find.
Here is a nice example: Denote the th prime by , and let . Then I claim that can be generated with elements. Indeed, take to be in each factor equal to some two generators of , and take to be an element of the form
where, for each , is any -cycle. Then the th power of is a -cycle in the th factor. With and we can generate all the conjugates of this cycle, and these together generate the whole th factor by simplicity.