# Difference between revisions of "ApCoCoA-1:Other12 groups"

### Other groups

#### Description

This group has the following finite representation:

```G = <x,t | tx^{a}t^{-1} = x^{b},t^{n} = 1>
```

for a,b >= 1 and n >= 2.

#### 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.N := 4;

// x is invers to z, t has an implicit invers (Relation: t^{n} = 1)
Use ZZ/(2)[x,t,z];
NC.SetOrdering("LLEX");

Define CreateRelationsOther12()
Relations:=[];

// add the invers relations xz = zx = 1
Append(Relations,[[x,z],]);
Append(Relations,[[z,x],]);

// add the relation t^{n} = 1
RelationBuffer0:=[];
For Index0:=1 To MEMORY.N Do
Append(RelationBuffer0,t);
EndFor;
Append(Relations,[RelationBuffer0,]);

// add the relation tx^{a}t^{-1} = x^{b}
RelationBuffer1:=[];
Append(RelationBuffer1,t);
Append(RelationBuffer1,x^(MEMORY.A));
Append(RelationBuffer1,t^(MEMORY.N-1));
Append(Relations,[RelationBuffer1,[x^MEMORY.B]]);

Return Relations;
EndDefine;

Relations:=CreateRelationsOther12();
Gb:=NC.GB(Relations,31,1,100,1000);
```

#### Examples in Symbolic Data Format

##### Other group 12 a=1 b=1 n=2
``` <FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier">
<vars>t,x,z</vars>
<basis>
<ncpoly>x*z-1</ncpoly>
<ncpoly>z*x-1</ncpoly>
<ncpoly>(t^2)-1</ncpoly>
<ncpoly>t*(x^1)*(t^(2-1))-(x^1)</ncpoly>
</basis>
<Comment>Other_groups_12a1b1n2</Comment>
</FREEALGEBRA>
```
##### Other group 12 a=3 b=3 n=4
``` <FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier">
<vars>t,x,z</vars>
<uptoDeg>10</uptoDeg>
<basis>
<ncpoly>x*z-1</ncpoly>
<ncpoly>z*x-1</ncpoly>
<ncpoly>(t^4)-1</ncpoly>
<ncpoly>t*(x^3)*(t^(4-1))-(x^3)</ncpoly>
</basis>
<Comment>The partial LLex Gb has 106 elements</Comment>
<Comment>Other_groups_12a3b3n4</Comment>
</FREEALGEBRA>
```
##### Other group 12 a=4 b=4 n=4
``` <FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier">
<vars>t,x,z</vars>
<uptoDeg>11</uptoDeg>
<basis>
<ncpoly>x*z-1</ncpoly>
<ncpoly>z*x-1</ncpoly>
<ncpoly>(t^4)-1</ncpoly>
<ncpoly>t*(x^4)*(t^(4-1))-(x^4)</ncpoly>
</basis>
<Comment>The partial LLex Gb has 182 elements</Comment>
<Comment>Other_groups_12a4b4n4</Comment>
</FREEALGEBRA>
```