The Heisenberg group: the group of 3 by 3 upper triangular matrices of the form

1 a c

0 1 b

0 0 1

When making a first step to non-commutativity? Or beyond planar models? Try the Heisenberg group.

The Heisenberg group: the group of 3 by 3 upper triangular matrices of the form

$$\begin{pmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}.$$

When making a first step to non-commutativity? Or beyond planar models? Try the Heisenberg group.