Classification of Bieberbach groups Does anybody know if there exists a list of the four dimensional Bieberbach groups presented by generators and relations on the web?. I know there exists the book Crystallographic Groups of Four-Dimensional Space by Brown, Bulow and Neubuser, but I do not have it at the moment.
 A: You may want to look into GAP, a system for computational discrete algebra, which contains the CrystCat package. This package contains all information contained in the book you mention. 
To check if a group is Bieberbach, you can use DisplaySpaceGroupType, which will mention "fp-free" if the group is torsion-free. 
gap> DisplaySpaceGroupType(3,6,1,1,1);
#I     Space-group type (3,6,1,1,1); IT(168) = P6; orbit size 1
gap> DisplaySpaceGroupType(3,6,1,1,4);
#I    *Space-group type (3,6,1,1,4); IT(169) = P61, IT(170) = P65;
#I      orbit size 2; fp-free

If you want a simple list of generators and relations, see the following example:
gap> S := SpaceGroupOnRightBBNWZ(3,6,1,1,4);
SpaceGroupOnRightBBNWZ( 3, 6, 1, 1, 4 )
gap> F := Image(IsomorphismFpGroup(S));
<fp group on the generators [ f1, f2, f3, f4, f5 ]>
gap> RelatorsOfFpGroup(F);
[ f1^3*f5^2, f2^-1*f1^-1*f2*f1, f2^2*f5, f3^-1*f1^-1*f3*f1*f3*f4^-1,
  f4^-1*f1^-1*f4*f1*f3*f4^2, f5^-1*f1^-1*f5*f1, f3^-1*f2^-1*f3*f2*f3^2,
  f4^-1*f2^-1*f4*f2*f4^2, f5^-1*f2^-1*f5*f2, f4^-1*f3^-1*f4*f3,
  f5^-1*f3^-1*f5*f3, f5^-1*f4^-1*f5*f4 ]

You can then simplify this presentation as follows:
gap> G := SimplifiedFpGroup(F);
<fp group on the generators [ f1, f2, f4 ]>
gap> RelatorsOfFpGroup(G);
[ f2^-1*f1^-1*f2*f1, f2^-1*f4*f2*f4, f2^-4*f1^3, (f4^-1*f1)^2*f4^-1*f1^-2, 
  f1^-1*(f4*f1)^2*f4*f1^-1, f2*f4^-1*f1^-1*f2^-2*f1*f4*f2 ]

You may also be interested in the Carat package, which allows you to calculate a database of higher-dimensional crystallographic groups and the AcLib package, which contains a database of almost-crystallographic groups.
A: See Sections 3 and 4 of Chapter 8 of my book
``Four-Manifolds, Geometries and Knots",
Geometry and Topology Monographs 5 (2002)
(revised in 2007 and 2014).
Presentations are given for the groups which are not {\it uniquely}
semidirect products $G\rtimes_\theta\mathbb{Z}$. 
In the remaining cases, $G$ is a flat
3-manifold group, and the possible automorphisms $\theta$ are given.
Presentations and automorphisms of flat 3-manifold groups
are considered in Section 2 of the same chapter, 
and it is straightforward to write down a presentation
for each such semidirect product.
The list of 4-dimensional Bieberbach groups is also rederived in 
the arXiv paper 1303.6613 [math.GT] by Lambert, Ratcliffe and Tschantz.
(They noticed an oversight in my treatment of the four examples
with finite abelianization, but the presentations are correct.) 
