# ApCoCoA-1:OrdinaryTetrahedron groups

### Ordinary Tetrahedron Groups

#### Description

An Ordinary Tetrahedron group is a group with a representation of the form

``` G(e_1,e_2,e_3,f_1,f_2,f_3) = <x,y,z | x^{e_1} = y^{e_2} = z^{e_3} = (xy^{-1})^{f_1} = (yz^{-1})^{f_2} = (zx^{-1})^{f_3} = 1>
```

#### Reference

Edjvet, Rosenberger, Stille, Thomas, "On certain generalized tetrahedon groups", Computational And Geometric Aspects Of Modern Algebra.

#### Computation

``` /*Use the ApCoCoA package ncpoly.*/

// Variables of Ordinary Tetrahedon group
MEMORY.E1:=3;
MEMORY.E2:=3;
MEMORY.E3:=3;
MEMORY.F1:=3;
MEMORY.F2:=3;
MEMORY.F3:=3;

Use ZZ/(2)[x,y,z];
NC.SetOrdering("LLEX");

Define CreateRelationsDicyclic()
Relations:=[];
If MEMORY.E1 < 2 Or MEMORY.E2 < 2 Or MEMORY.E3 < 2 Or MEMORY.F1 < 2 Or MEMORY.F2 < 2 Or MEMORY.F3 < 2 Then
Output:="Wrong Input! Please check that E_i and F_i are greater than 1";
Print(Output);
Else
// add the relations x^{e_1} = 1, y^{e_2} = 1 and z^{e_3} = 1
Append(Relations,[[x^(MEMORY.E1)],[1]]);
Append(Relations,[[y^(MEMORY.E2)],[1]]);
Append(Relations,[[z^(MEMORY.E3)],[1]]);

RelationA:=[];
For Index1 := 1 To MEMORY.F1 Do
Append(RelationA,x);
Append(RelationA,y^(MEMORY.E2-1));
EndFor;
Append(Relations,[RelationA,[1]]);

RelationB:=[];
For Index2 := 1 To MEMORY.F2 Do
Append(RelationB,y);
Append(RelationB,z^(MEMORY.E3-1));
EndFor;
Append(Relations,[RelationB,[1]]);

RelationC:=[];
For Index3 := 1 To MEMORY.F3 Do
Append(RelationC,z);
Append(RelationC,x^(MEMORY.E1-1));
EndFor;
Append(Relations,[RelationC,[1]]);
EndIf;

Return Relations;
EndDefine;

Relations:=CreateRelationsDicyclic();
Relations;

If Size(Relations) > 0 Then
Gb:=NC.GB(Relations,31,1,100,1000);
Size(Gb);
EndIf;
```

#### Example in Symbolic Data Format

``` <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier">
<vars>x,y,z</vars>
<uptoDeg>15</uptoDeg>
<basis>
<ncpoly>x^3-1</ncpoly>
<ncpoly>y^3-1</ncpoly>
<ncpoly>z^3-1</ncpoly>
<ncpoly>((x*y^(3-1))^(3))-1</ncpoly>
<ncpoly>((y*z^(3-1))^(3))-1</ncpoly>
<ncpoly>((z*x^(3-1))^(3))-1</ncpoly>
</basis>
<Comment>The partial LLex Gb has 140 elements</Comment>
<Comment>Ordinary_Tetrahedron_group_3</Comment>
</FREEALGEBRA>
```