predefined by FR package of GAP CAS. Here BinaryKneadingGroup("1") is BasilicaGroup.
gap> BinaryKneadingGroup(1/3)=BasilicaGroup;
true
or :
gap> B := FRGroup("a=<1,b>(1,2)","b=<1,a>",IsFRMealyElement);
<state-closed group over [ 1, 2 ] with 2 generators>
gap> AssignGeneratorVariables(B);
#I Assigned the global variables [ "a", "b" ]
gap> B=BasilicaGroup;
#I \=: converting second argument to FR element
#I \<: converting second argument to FR element
#I \<: converting second argument to FR element
#I \=: converting second argument to FR element
#I \=: converting second argument to FR element
#I \<: converting second argument to FR element
#I \<: converting second argument to FR element
#I \=: converting second argument to FR element
#I \=: converting first argument to FR element
#I \=: converting first argument to FR element
#I \=: converting first argument to FR element
#I \=: converting first argument to FR element
#I \=: converting first argument to FR element
#I \=: converting first argument to FR element
#I \=: converting first argument to FR element
#I \=: converting first argument to FR element
true
↑R. I. Grigorchuk and A. Zuk (2002a). On a torsion-free weakly branch group defined by
a three state automaton. Internat. J. Algebra Comput., 12(1-2):223–246. International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup
Theory (Lincoln, NE, 2000).