Binary von Dyck group

Definition
The binary von Dyck group or binary triangle group with parameters $$(p,q,r)$$ is defined as follows:

$$\langle a,b,c \mid a^p = b^q = c^r = abc \rangle$$.

This group is sometimes denoted $$\Gamma(p,q,r)$$.

In the particular case where $$q = r = 2$$ or $$(p,q,r) = (2,3,5),(2,3,4),(2,3,3)$$, the center of the group, which is generated by $$abc$$, has order two, and the quotient is the von Dyck group. For other values of $$(p,q,r)$$, the center may not have finite order.

Relation with other groups
The quotient of the binary von Dyck group by the subgroup generated by $$abc$$ is the von Dyck group with parameters $$(p,q,r)$$. In many cases of interest, the element $$abc$$ has order two and the subgroup generated by it is precisely the center of the binary von Dyck group.

Each of the von Dyck groups that arise as finite subgroups of the special orthogonal group $$SO(3,\R)$$ have corresponding binary von Dyck groups of interest:


 * The dihedral groups, which are von Dyck groups for parameters $$(2,2,n)$$, correspond to the binary dihedral groups, also called the dicyclic groups.
 * The tetrahedral group, which is the alternating group of degree four and is von Dyck for parameters $$(2,3,3)$$, corresponds to the binary tetrahedral group, which is isomorphic to special linear group:SL(2,3).
 * The octahedral group, which is the symmetric group of degree four, and is von Dyck for parameters $$(2,3,4)$$, corresponds to the binary octahedral group, which is a non-split central extension of a group of order two by the symmetric group of degree four. This is not isomorphic to general linear group:GL(2,3).
 * The icosahedral group, which is the alternating group of degree five, and is von Dyck for parameters $$(2,3,5)$$, corresponds to the binary icosahedral group, which is isomorphic to special linear group:SL(2,5).