The group in the problem is the group of units $(\mathbb{Z}/1000000\mathbb{Z})^\times$
The answer gives $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2^4\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/5^5\mathbb{Z}$.
I've been reading about decompositions of groups into other groups and I'm just completely stumped and confused by all the different decomposition theorems. Could someone help me explain this (right now, very arbitrary looking) decomposition into its factors and the relevant theorems being applied here?
It's just $$(\Bbb Z/2^6\Bbb Z)^\times \times(\Bbb Z/5^6\Bbb Z)^\times.$$ For odd $p$, $(\Bbb Z/p^m\Bbb Z)^\times$ is always cyclic. For $(\Bbb Z/2^m\Bbb Z)^\times$, see my previous reply.