Diffuse property of low dimensional Bieberbach groups

Diffuse property of low dimensional Bieberbach groups in CSV format and in a SQL like table.

The above files contain only the results. The gzipped gap file contains more details. The data in the file should be read as follows. Assume that \(\Gamma\) is a Bieberbach group of dimension \(n, n\leq 6\) that fits into the following short exact sequence \[ 0 \longrightarrow M \longrightarrow \Gamma \longrightarrow G \longrightarrow 1, \] where \(M=\mathbb{Z}^n\) is the maximal abelian normal subgroup of \(\Gamma\) and \(G\) is the holonomy group of \(\Gamma\). If the rank of the center of \(\Gamma\) is equal to \(k\), then we have an epimorphism \[ f \colon \Gamma \to \mathbb{Z}^k. \] Note that in this case \(\Gamma \subset \operatorname{GL}(n,\mathbb{Z}) \ltimes \mathbb{Q}^n \subset \operatorname{GL}(n+1,\mathbb{Q})\), up to conjugation in \(\operatorname{GL}(n+1,\mathbb{Q})\), contains of matrices of the form \[ \begin{bmatrix} * & * & *\\ 0 & I_k & *\\ 0 & 0 & 1\end{bmatrix} \] and the following map from \(\ker f\) to \(\operatorname{GL}(n-k+1,\mathbb{Q})\) \[ \begin{bmatrix} A & * & a\\ 0 & I_k & 0\\ 0 & 0 & 1\end{bmatrix} \mapsto \begin{bmatrix} A & a\\0 & 1\end{bmatrix} \] is an isomorphism.

Description of fields of GAP record is given in the following table.

nameCARAT name of the group
dimdimension \(n\) of \(\Gamma\)
cgensgenerators of the group (without lattice generators)
holonomyGAP id of \(G\)
hdiffis every Sylow subgroup of \(G\) cyclic (and hence \(\Gamma\) is diffuse group)
zrankrank of the center of \(\Gamma\)
sub.complementbasis of \(M^G \cong Z(\Gamma)\)
sub.subcomplement of sub.complement to a basis of \(M\)
kgensgenerators of \(\operatorname{ker}f\) as \(n-k\)-dimensional Bieberbach group (without lattice generators)
knameCARAT name of \(\operatorname{ker}f\)
diffuseinformation about diffuse property of \(\Gamma\)