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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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Generators, general antiburgers: X Y T;
Weights:                         1 1 1;
Limiting weight: 6;
Parameters: p_1 p_2 p_3 p_4 p_5;
Relations:
[[Y,X],X] + p_1 [Y,X] - p_3 Y;
[[Y,X],Y] + [X,T] +  p_4 Y;
p_2 [Y,X] + [T,Y] + p_5 Y;