# ApCoCoA-1:Extended Hecke groups

### Extended Hecke Groups

#### Reference

Ma. Louise Antonette N. De Las Peñas, Ma. Carlota B. Decena and Glenn R. Laigo, ON THE SUBGROUPS OF THE EXTENDED HECKE, HECKE AND PICARD GROUPS, Proceedings of the 5th Asian Mathematical Conference, Malaysia 2009.

#### Computation

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

// Define the variable p of the extended Hecke group
MEMORY.P := 3;

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

Define CreateRelationsExtendedHeckegroup()
Relations:=[];

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

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

//add the relation y^p = 1
Append(Relations,[[y^MEMORY.P],[1]]);

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

// add the relation (yr)^2 = 1
Append(Relations,[[y,r,y,r],[1]]);
Return Relations;
EndDefine;

Relations:=CreateRelationsExtendedHeckegroup();
Relations;

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

#### Example in Symbolic Data Format

```<FREEALGEBRA createdAt="2014-07-03" createdBy="strohmeier">
<vars>r,x,y</vars>
<uptoDeg>53</uptoDeg>
<basis>
<ncpoly>r*r-1</ncpoly>
<ncpoly>x*x-1</ncpoly>
<ncpoly>y^3-1</ncpoly>
<ncpoly>r*x*r*x-1</ncpoly>
<ncpoly>y*r*y*r-1</ncpoly>
</basis>
<Comment>The partial LLex Gb has 69 elements</Comment>
<Comment>Extended_Hecke_Group_p3</Comment>
</FREEALGEBRA>
```