The strange result given by IsSymmorphicSpaceGroup.

[hongyi-zhao]

See my following testing:

gap> SGGenSet229me;
[ [ [ 0, 0, -1, 1/2 ], [ -1, 0, 0, 1/2 ], [ 0, -1, 0, 1/2 ], [ 0, 0, 0, 1 ] ], [ [ 55/8, -33/8, -25/8, 7/32 ], [ 29/4, -15/4, -15/4, 1/16 ], [ 55/8, -25/8, -33/8, 23/32 ],
      [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ] ]
gap> S2:=AffineCrystGroupOnLeft( SGGenSet229me );
<matrix group with 3 generators>
gap> S3:=SpaceGroupOnLeftIT(3,229);
SpaceGroupOnLeftIT(3,229,'1')
gap> IsSymmorphicSpaceGroup(S2);
false
gap> IsSymmorphicSpaceGroup(S3);
true
gap> IdentifySpaceGroup(S2);
[ [ 229, [ "max.5", "3 1", "0" ], "I 4/m -3 2/m" ], [ [ -5/8, 5/8, 3/4, -3/4 ], [ -5/8, 3/4, 5/8, -3/4 ], [ -3/4, 5/8, 5/8, -3/4 ], [ 0, 0, 0, 1 ] ] ]
gap> IdentifySpaceGroup(S3);
[ [ 229, [ "max.5", "3 1", "0" ], "I 4/m -3 2/m" ], [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ] ]
gap> IsSymmorphicSpaceGroup(StandardAffineCrystGroup(S2));
false
gap> conj:=ConjugatorSpaceGroups(S2, S3);
[ [ -11/2, 5/2, 5/2, 5/8 ], [ 5/2, -11/2, 5/2, 5/8 ], [ -5/2, -5/2, 11/2, 11/8 ], [ 0, 0, 0, 1 ] ]
gap> S2^conj=S3;
true
gap> IsSymmorphicSpaceGroup( S2^conj );
false
gap> IsSymmorphicSpaceGroup( S3 );
true

As you can see, S2 and S3 are isomorphic, but only S3 is recognized as symmorphic, and even S2^conj, which is identical to S3, cannot be recognized as symmorphic.

Regards,
Zhao