# ApCoCoA-1:Other13" groups

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### Other groups

#### Description

This group has the following finite representation:

G = <x,y | x^2 = xy^{a}xy^{b}xy^{c}xy^{d} = 1>

#### Reference

No reference available

#### Computation

/*Use the ApCoCoA package ncpoly.*/

// Note that a,b >= 1 and n >= 2
MEMORY.A := 3;
MEMORY.B := 3;
MEMORY.C := 4;
MEMORY.D := 5;
// y is invers to z, the invers element of x follows directly from the relation x^2 = 1
Use ZZ/(2)[x,y,z];
NC.SetOrdering("LLEX");
Define CreateRelationsOther13()
Relations:=[];

// add the relation of the invers elements yz = zy = 1
Append(Relations,[[y,z],[1]]);
Append(Relations,[[z,y],[1]]);

// add the relation x^2 = 1
Append(Relations,[[x,x],[1]]);

// add the relation xy^{a}xy^{b}xy^{c}xy^{d}
Append(Relations,[[x,y^(MEMORY.A),x,y^(MEMORY.B),x,y^(MEMORY.C),x,y^(MEMORY.D)],[1]]);
Return Relations;
EndDefine;

Relations:=CreateRelationsOther13();
GB:=NC.GB(Relations,31,1,100,1000);