I'm struggling with this question: find two elements of the group G = Z2 x Z3 x Z4 that generate the group. In Z2 x Z3, which are coprime, (1,1) is a generator with order 6, but this doesn't extend then to (1,1,1) as a generator of G, because it has order 12, not 24 as required, so I can't even find one generator! Can anyone offer any advice? I know how to solve this problem if the numbers were coprime but that's not the case here.
Thanks for any help.