# ApCoCoA-1:Other2 groups

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### Other groups

#### Description

The first group is called Rosenberger-Monster and is the largest finite generalized triangle group. A finite representation of G is given below:

G = <a,b | a^2 = b^3 = (abababab^{2}ab^{2}abab^{2}ab^{2})^2 = 1>

The second group is already infinite and denoted by H:

H = <a,b | a^2 = b^3 = (abababab^{2}abab^{2}ab^{2})^2 = 1>

#### Reference

On the Rosenberger Monster Robert Fitzgerald Morse, Department of Electrical Engineering and Computer Science, University of Evansville IN 47722 USA

#### Computation of G

/*Use the ApCoCoA package ncpoly.*/

Use ZZ/(2)[a,b];
NC.SetOrdering("LLEX");
Define CreateRelationsOther2()
Relations:=[];

// add the relations a^2 = b^3 = 1
Append(Relations,[[a,a],[1]]);
Append(Relations,[[b,b,b],[1]]);
// add the relation (abababab^{2}ab^{2}abab^{2}ab^{2})^2 = 1
Append(Relations,[[a,b,a,b,a,b,a,b,b,a,b,b,a,b,a,b,b,a,b,b,a,b,a,b,a,b,a,b,b,a,b,b,a,b,a,b,b,a,b,b],[1]]);
Return Relations;
EndDefine;

Relations:=CreateRelationsOther2();
GB:=NC.GB(Relations,31,1,100,1000);

#### Computation of H

/*Use the ApCoCoA package ncpoly.*/

Use ZZ/(2)[a,b];
NC.SetOrdering("LLEX");
Define CreateRelationsOther3()
Relations:=[];

// add the relations a^2 = b^3 = 1
Append(Relations,[[a,a],[1]]);
Append(Relations,[[b,b,b],[1]]);
// add the relation (abababab^{2}abab^{2}ab^{2})^2 = 1
Append(Relations,[[a,b,a,b,a,b,a,b,b,a,b,a,b,b,a,b,b,a,b,a,b,a,b,a,b,b,a,b,a,b,b,a,b,b],[1]]);
Return Relations;
EndDefine;

Relations:=CreateRelationsOther3();
GB:=NC.GB(Relations,31,1,100,1000);