# 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],[1]]);
Append(Relations,[[z,x],[1]]);

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

// 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);
```