How do I construct a nonabelian group of order 1575?

Question

I think that it should be a semidirect product of the direct product of any two of the three groups $\mathbb Z/7\mathbb Z$, $\mathbb Z/9\mathbb Z$ and $\mathbb Z/25\mathbb Z$ and the other one. But which of these will be the normal subgroup and what nontrivial homomorphism should I take? Can someone please give a solution?

Answer 1

There's a unique nonabelian group of order 21, the semidirect product of $\Bbb Z_7$ and $\Bbb Z_3$. Note that $\text{Aut}(\Bbb Z_7) \cong \Bbb Z_6$; take your map to be $\Bbb Z_3 \xrightarrow{\times 2} \Bbb Z_6$. Now take the product of this with, say, $\Bbb Z_{75}$.