Consider the multiplicative group generated by matrices of the form $$ \begin{bmatrix} {1} & { 0} & { c_1} & {c_3} \\ {0} & {1} & {c_3} & {c_2} \\ {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {1} \end{bmatrix} $$
along with their transposes $$ \begin{bmatrix} {1} & { 0} & {0} & {0} \\ {0} & {1} & {0} & {0} \\ {c_4} & {c_6} & {1} & {0} \\ {c_6} & {c_5} & {0} & {1} \end{bmatrix} $$ where $c_i \in \mathbb{R}.$ Is there any easier way to describe this group, e.g. by identifying it with one of the more typical matrix groups?
More generally we can consider groups generated by matrices of the form $$ \begin{bmatrix} {I_n} & {C}\\ {0} & {I_n}\\ \end{bmatrix} $$ and their transposes, where $C$ is symmetric. Is there any better way to describe these groups?
