Take the set $\{\pm1\}$ with multiplication as a model for $\Z_2$ and define $F:\Z_2^3\rar\OO(3)$ by: $$ \forall \e=(\e_1,\e_2,\e_3)\in\Z_2^3, \forall x\in\R^3:\quad F(\e)e_j\colon=\e_j e_j $$ where $e_1,e_2,e_3$ denotes the standard basis of $\R^3$. Then $F$ is a homomorphism and since $(-1,1,1)$, $(1,-1,1)$, $(1,1,-1)$ generate $\Z_2^3$ and $F(-1,1,1)$, $F(1,-1,1)$, $F(1,1,-1)$ are just the reflections about the coordinate planes, we conclude that: $F(\Z_2^3)$ coincides with the group $G$ generated by the reflections about the coordinate planes, i.e. $F:\Z_2^3\rar G$ is an isomorphism.