Consider the following group presentation:

where $l,m,n\in\mathbb{N}$.

A group with this presentation corresponds to a triangle; roughly, the generators are reflections in its sides and its angles are $\pi/l,\pi/m,\pi/n$.

Denote by $D(l,m,n)$ the subgroup of index 2 in $\Delta(l,m,n)$, corresponding to preservation of orientation of the triangle.

The $D(l,m,n)$ are defined by the following presentation:

Note that $D(l,m,n)\cong D(m,l,n)\cong D(n,m,l)$, so $D(l,m,n)$ is independent of the order of the $l,m,n$.

Arising from the geometrical nature of these groups,

is called the spherical case,

is called the Euclidean case, and

is called the hyperbolic case

Groups either of the form $\Delta(l,m,n)$ or $D(l,m,n)$ are referred to as triangle groups; groups of the form $D(l,m,n)$ are sometimes refered to as von Dyck groups.