Let $G = \left\{\begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \textrm{ with } a=\pm 1, b \in \mathbb{Z}, c= \pm 1 \right\} \subset GL_2(\mathbb{R})$
and $H = \left\{\begin{pmatrix} a & b \\ 0 & 1\end{pmatrix} \in G \right\}$.
Show or disprove that $G$ is isomorphic to $H \times \{\pm 1\}$.
I tried constructing the obvious mappings $G \to H \times \{\pm 1\}$ but they turned out to be non-homomorphic. I then tried to disprove the conjecture by looking at the orders of elements in $G$ and $H \times \{\pm 1\}$, but they both turned out to posses only countable infinite elements of order 2 and $\infty$. Furthermore both $G$ and $H \times \{\pm 1\}$ are Abelian.
Now I'm stuck. In the preceding exercise I have proven $G/[G,G] \simeq (\mathbb{Z}/2\mathbb{Z})^3$ so maybe this can be used?