gap> LoadPackage( "UnitalSZ", false ); gap> LoadPackage( "grape", false ); gap> u:=KNPAbstractUnital(1577); KNPAbstractUnital(1577) gap> AutomorphismGroup(u); <permutation group with 6 generators> gap> b:=BlocksOfUnital(u)[1]; [ 1, 2, 5, 6, 14 ] gap> bls0:=Filtered(BlocksOfUnital(u),x->Size(Intersection(x,b))=1);; gap> bls:=List(bls0,x->Difference(x,b));; gap> cols:=AllRegularBlockColorings(bls,5,Group(())); time; [ Transformation( [ 1, 1, 1, 4, 2, 5, 3, 5, 3, 4, 2, 2, 5, 3, 4, 4, 3, 2, 5, 5, 3, 4, 4, 5, 3, 2, 2, 1, 1, 1, 5, 3, 2, 4, 5, 2, 5, 4, 3, 2, 2, 3, 2, 5, 3, 4, 1, 1, 1, 4, 2, 4, 4, 5, 2, 3, 2, 5, 3, 1, 1, 1, 3, 3, 2, 5, 3, 5, 4, 4, 4, 5, 1, 1, 1 ] ), Transformation( [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 3, 4, 2, 4, 2, 3, 3, 2, 4, 2, 5, 5, 5, 5, 3, 2, 2, 4, 5, 2, 5, 3, 5, 3, 2, 2, 3, 5, 3, 4, 4, 4, 4, 3, 5, 3, 5, 4, 4, 5, 3, 5, 2, 2, 2, 5, 2, 4, 2, 4, 5, 5, 4, 2, 4, 3, 3, 3 ] ), Transformation( [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 5, 5, 5, 3, 3, 3, 5, 3, 3, 5, 4, 5, 3, 2, 2, 5, 3, 2, 5, 2, 4, 2, 5, 5, 4, 4, 4, 4, 2, 5, 5, 3, 3, 2, 3, 2, 2, 5, 2, 5, 5, 3, 2, 3, 2, 4, 3, 4, 4, 4, 2, 3, 2, 4, 2, 3 ] ), Transformation( [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 3, 2, 5, 3, 2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ] ), Transformation( [ 1, 2, 5, 5, 5, 5, 5, 2, 1, 2, 2, 1, 1, 2, 1, 4, 4, 4, 4, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 4, 5, 3, 2, 1, 3, 5, 1, 5, 1, 4, 1, 5, 5, 4, 4, 4, 4, 1, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 2, 4, 5, 4, 4, 4, 2, 5, 2, 4, 2, 5 ] ), Transformation( [ 1, 3, 4, 4, 4, 4, 4, 3, 1, 3, 3, 1, 1, 3, 1, 4, 4, 4, 4, 5, 5, 5, 1, 1, 1, 5, 1, 1, 5, 4, 5, 3, 2, 1, 5, 2, 1, 5, 1, 3, 1, 5, 5, 3, 3, 3, 3, 1, 5, 5, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 3, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ] ), Transformation( [ 1, 2, 4, 4, 4, 4, 4, 2, 1, 2, 2, 1, 1, 2, 1, 4, 4, 4, 4, 2, 2, 2, 3, 3, 3, 2, 3, 3, 2, 4, 5, 3, 2, 5, 1, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 1, 3, 3, 4, 3, 4, 4, 1, 4, 1, 1, 3, 4, 3, 2, 1, 3, 1, 1, 1, 2, 3, 2, 1, 2, 3 ] ), Transformation( [ 1, 3, 5, 5, 5, 5, 5, 3, 1, 3, 3, 1, 1, 3, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 3, 2, 2, 1, 5, 2, 5, 2, 3, 2, 5, 5, 3, 3, 3, 3, 2, 5, 1, 3, 3, 2, 3, 2, 2, 1, 2, 1, 1, 3, 2, 5, 2, 1, 5, 1, 1, 1, 2, 5, 2, 1, 2, 5 ] ), Transformation( [ 1, 5, 2, 1, 5, 1, 5, 2, 2, 2, 1, 2, 5, 1, 5, 4, 1, 1, 4, 1, 4, 1, 3, 3, 3, 4, 3, 3, 5, 2, 5, 3, 2, 2, 4, 3, 2, 5, 5, 2, 5, 1, 1, 1, 2, 1, 4, 4, 4, 4, 3, 3, 5, 3, 4, 4, 5, 5, 5, 1, 3, 2, 3, 2, 4, 3, 4, 2, 2, 4, 3, 4, 1, 5, 3 ] ), Transformation( [ 1, 4, 3, 1, 4, 1, 4, 3, 3, 3, 1, 3, 4, 1, 4, 4, 1, 1, 4, 1, 3, 1, 3, 3, 4, 3, 4, 5, 5, 5, 5, 3, 2, 2, 3, 5, 2, 5, 2, 5, 2, 1, 1, 1, 5, 1, 4, 2, 3, 3, 3, 5, 2, 5, 2, 2, 5, 2, 5, 1, 4, 2, 5, 2, 4, 4, 4, 5, 5, 2, 4, 2, 1, 2, 3 ] ), Transformation( [ 1, 4, 2, 1, 4, 1, 4, 2, 2, 2, 1, 2, 4, 1, 4, 4, 3, 3, 4, 5, 5, 5, 3, 3, 4, 5, 4, 1, 5, 2, 5, 3, 2, 2, 5, 1, 2, 5, 3, 2, 3, 5, 5, 3, 2, 3, 4, 1, 5, 5, 3, 1, 3, 1, 1, 1, 5, 3, 5, 5, 4, 2, 1, 2, 4, 4, 4, 2, 2, 1, 4, 1, 3, 3, 3 ] ), Transformation( [ 1, 5, 3, 1, 5, 1, 5, 3, 3, 3, 1, 3, 5, 1, 5, 4, 4, 4, 4, 2, 3, 2, 3, 3, 2, 3, 2, 1, 5, 4, 5, 3, 2, 2, 3, 1, 2, 5, 5, 4, 5, 2, 2, 4, 4, 4, 4, 1, 3, 3, 3, 1, 5, 1, 1, 1, 5, 5, 5, 2, 2, 2, 1, 2, 4, 2, 4, 4, 4, 1, 2, 1, 4, 5, 3 ] ) ] 342 gap> List(cols,c->ParamodificationOfUnitalNC(u,b,c)); time; [ AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4> ] 32 gap> List(cols,c->ParamodificationOfUnital(u,b,c)); time; [ AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4> ] 88 gap> ParamodificationsOfUnitalWithBlock(u,b); time; [ AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4> ] 219 gap> AllParamodificationsOfUnital(u); time; [ AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4>, AU_UnitalDesign<4> ] 961
We say that a unital is para-rigid, if all block colorings of \(C(b)\) are equivalent with the trivial one. The following example shows that the cyclic unitals of order 4 and 6 by Bagchi and Bagchi are para-rigid.
gap> SetInfoLevel(InfoParamod,2); gap> u:=BagchiBagchiCyclicUnital(4); BagchiBagchiCyclicUnital(4) gap> AllParamodificationsOfUnital(u); #I 2 block representatives for the unital computed... #I 1 coloring(s) for the given unital-block pair computed... #I 1 coloring(s) for the given unital-block pair computed... [ ] gap> u:=BagchiBagchiCyclicUnital(6); BagchiBagchiCyclicUnital(6) gap> AllParamodificationsOfUnital(u); #I 2 block representatives for the unital computed... #I 1 coloring(s) for the given unital-block pair computed... #I 1 coloring(s) for the given unital-block pair computed... [ ]
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