# Difference between revisions of "ApCoCoA-1:Other5 groups"

### Other groups

#### Description

The first group, denoted by G, has an order |G| = 4224 and can be represented as:

``` G = <a,b | a^{2}b^{-4} = (ababab^{3})^{2} = 1>
```

The second group, denoted by H, is also solvable and has the following representation:

``` H = <a,b | a^{2}b^{4} = (ababab^{3})^{2} = 1>
```

#### Reference

No reference available

#### Computation of G

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

// a is invers to c and b is invers to d
Use ZZ/(2)[a,b,c,d];
NC.SetOrdering("LLEX");
Define CreateRelationsOther5()
Relations:=[];

// add the invers relations ac = ca = bd = db = 1
Append(Relations,[[a,c],]);
Append(Relations,[[c,a],]);
Append(Relations,[[b,d],]);
Append(Relations,[[d,b],]);

// add the relation a^{2}b^{-4} = 1
Append(Relations,[[a,a,d,d,d,d],]);

// add the relation (ababab^{3})^{2} = 1
Append(Relations,[[a,b,a,b,a,b,b,b,a,b,a,b,a,b,b,b],]);

Return Relations;
EndDefine;

Relations:=CreateRelationsOther5();
GB:=NC.GB(Relations,31,1,100,1000);
```

#### Computation of H

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

// a is invers to c and b is invers to d
Use ZZ/(2)[a,b,c,d];
NC.SetOrdering("LLEX");
Define CreateRelationsOther6()
Relations:=[];

// add the invers relations ac = ca = bd = db = 1
Append(Relations,[[a,c],]);
Append(Relations,[[c,a],]);
Append(Relations,[[b,d],]);
Append(Relations,[[d,b],]);

// add the relation a^{2}b^{4} = 1
Append(Relations,[[a,a,b,b,b,b],]);

// add the relation (ababab^{3})^{2} = 1
Append(Relations,[[a,b,a,b,a,b,b,b,a,b,a,b,a,b,b,b],]);

Return Relations;
EndDefine;

Relations:=CreateRelationsOther6();
GB:=NC.GB(Relations,31,1,100,1000);
```