Consider the following group presentation: $$\Delta(l,m,n)=\langle a,b,c:a^2,b^2,c^2,(ab)^l,(bc)^n,(ca)^m\rangle$$ 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: $$D(l,m,n)=\langle x,y:x^l,y^m,(xy)^n\rangle$$

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, $$1/l+1/m+1/n>1$$ is called the spherical case,$$1/l+1/m+1/n=1$$ is called the Euclidean case, and$$1/l+1/m+1/n<1$$ 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.