Five dimensional Bieberbach groups with trivial centre (Q1813319)

The author gives a list of all the finite holonomy groups of compact flat manifolds of dimension 5 with first Betti number 0. Up to affine equivalence, such manifolds are classified by their fundamental groups -- the torsion free Bieberbach groups. Along the way, the author raises the question: Let \(G\) be a finite 2-group with an elementary abelian subgroup of index 2 and order \(2^ k\) \((k>1)\). Assume \(G\) has a faithful integral representation of dimension \(k+2\). Is \(G\) a holonomy group of a \((k+2)\)- dimensional Bieberbach group \(B\)? Can \(B\) be taken to have first Betti number \(0\)? The assumption of having a faithful integral representation of dimension \(k+2\) is unnecessary. In fact, \(G\) already has a faithful integral representation of dimension \(k+1\). Namely, it is not difficult to see that \(G\) is isomorphic to a direct product of \(c\) cyclic groups of order 2 and a ``co-product'' of \(e\) cyclic group of order 4 (\(e = 0\) or 1) and \(d\) dihedral groups of order 8 over a cyclic group of order 2, where \(c + e + 2d = k\). There is then no problem showing the existence of a faithful integral representation of dimension \(k+1\).