# Difference between revisions of "ApCoCoA-1:Other13 groups"

### 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,c,d >= 1
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],]);
Append(Relations,[[z,y],]);

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

Append(Relations,[[x,y^(MEMORY.A),x,y^(MEMORY.B),x,y^(MEMORY.C),x,y^(MEMORY.D)],]);

Return Relations;
EndDefine;

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

#### Examples in Symbolic Data Format

##### Other group 13 a=2 b=3 c=3 d=4
``` <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
<vars>x,y,z</vars>
<uptoDeg>19</uptoDeg>
<basis>
<ncpoly>y*z-1</ncpoly>
<ncpoly>z*y-1</ncpoly>
<ncpoly>x*x-1</ncpoly>
<ncpoly>x*(y^2)*x*(y^3)*x*(y^3)*x*(y^4)-1</ncpoly>
<ncpoly></ncpoly>
</basis>
<Comment>The partial LLex Gb has 198 elements</Comment>
<Comment>Other_groups_13a2b3c3d4</Comment>
</FREEALGEBRA>
```
##### Other group 13 a=3 b=3 c=4 d=5
``` <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
<vars>x,y,z</vars>
<uptoDeg>18</uptoDeg>
<basis>
<ncpoly>y*z-1</ncpoly>
<ncpoly>z*y-1</ncpoly>
<ncpoly>x*x-1</ncpoly>
<ncpoly>x*(y^3)*x*(y^3)*x*(y^4)*x*(y^5)-1</ncpoly>
</basis>
<Comment>The partial LLex Gb has 4 elements</Comment>
<Comment>Other_groups_13a3b3c4d5</Comment>
</FREEALGEBRA>
```

##### Other group 13 a=5 b=2 c=4 d=3
``` <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
<vars>x,y,z</vars>
<uptoDeg>17</uptoDeg>
<basis>
<ncpoly>y*z-1</ncpoly>
<ncpoly>z*y-1</ncpoly>
<ncpoly>x*x-1</ncpoly>
<ncpoly>x*(y^5)*x*(y^2)*x*(y^4)*x*(y^3)-1</ncpoly>
</basis>
<Comment>The partial LLex Gb has 4 elements</Comment>
<Comment>Other_groups_13a5b2c4d3</Comment>
</FREEALGEBRA>
```