Find a subgroup of order $6$ in $U(450)$
I tried expressing it as an external direct product of cyclic groups of the form $Z_{n}$ but since the answer is given as $U_{50}(450)$, I figured it was the wrong approach.But I didn't understand how they arrived at this solution. I know that $$U_{50}(450)=U_{50}(50.9)\thickapprox U(9)$$ I'm stuck
Since $|U(450)|=\phi(450)=120$, the group $U(450)$ has an element $x$ of order $2$, and an element $y$ of order $3$, by Cauchy's Theorem. Hence $H:=\langle xy\rangle$ is a subgroup of order $6$.
For another solution see this duplicate question (replacing $700$ by $450$):
To Find a subgroup of order $6$ in $U(700).$