# ApCoCoA-1:Tetraeder group

### Tetraeder Group

#### Description

The Tetraeder group is isomorphic to the alternating group A_4. Geometrically this group is generated by the rotations of a regular tetrahedron. Its representation is given by

``` A_4 = <a,b | a^2 = b^3 = (ab)^3 = 1>
```

#### Reference

Geometries and Transformations, Manuscript, Chapter 11: Finite symmetry groups, N.W. Johnson, 2011

#### Computation

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

Use ZZ/(2)[a,b];
NC.SetOrdering("LLEX");

Define CreateRelationsTetraeder()
Relations:=[];
// add the relation a^2 = 1
Append(Relations,[[a^2],[1]]);

// add the relation b^3 = 1
Append(Relations,[[b^3],[1]]);

// add the relation (ab)^3 = 1
Append(Relations,[[a,b,a,b,a,b],[1]]);

Return Relations;
EndDefine;

Relations:=CreateRelationsTetraeder();
Gb:=NC.GB(Relations);
```

#### Example in Symbolic Data Format

``` <FREEALGEBRA createdAt="2014-01-24" createdBy="strohmeier">
<vars>a,b</vars>
<basis>
<ncpoly>a*a-1</ncpoly>
<ncpoly>b*b*b-1</ncpoly>
<ncpoly>(a*b)^3-1</ncpoly>
</basis>
<Comment>Tetraeder_group</Comment>
</FREEALGEBRA>
```