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Do the groups G and H given below have Z-equivalent copies which lie in a finite unimodular group?
G is generated by
6 0 0 0 0 0 -1 0 0 0 0 1 -1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0and H is generated by
6 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0
Bravais_grp g > gb
Bravais_grp h > hb
to write the Bravais groups of G and H into files 'gb' and
'hb'.
(Note, we have assumed already that G and H are finite. By calling
Is_finite g
Is_finite h
this could have been checked beforehand.)
Bravais_inclusions -S gb
Bravais_inclusions -S hb
to get lists of names for the Z-classes of all Bravais groups
containing G resp. H. We get
Bravais groups which contain a Z-equivalent subgroup
Symbol: 6-2' homogeneously d.: 1 zclass: 1
Symbol: 6-2 homogeneously d.: 1 zclass: 1
Symbol: 6-2 homogeneously d.: 2 zclass: 1
Symbol: 6-2 homogeneously d.: 3 zclass: 1
and
Bravais groups which contain a Z-equivalent subgroup
Symbol: 4-1';2-1 homogeneously d.: 1 zclass: 1
Symbol: 4-1;2-1 homogeneously d.: 1 zclass: 1
Symbol: 4-1;2-1 homogeneously d.: 2 zclass: 1
Symbol: 6-1 homogeneously d.: 1 zclass: 1
There are no common names. Hence there is no finite subgroup of GL_6(Z) containig G and a
GL6(Z)-conjugate of H.
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