# 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],[1]]);
Append(Relations,[[c,a],[1]]);
Append(Relations,[[b,d],[1]]);
Append(Relations,[[d,b],[1]]);

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

// 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],[1]]);

Return Relations;
EndDefine;

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

#### G in Symbolic Data Format

``` <FREEALGEBRA createdAt="2014-01-24" createdBy="strohmeier">
<vars>a,b,c,d</vars>
<uptoDeg>12</uptoDeg>
<basis>
<ncpoly>a*c-1</ncpoly>
<ncpoly>c*a-1</ncpoly>
<ncpoly>b*d-1</ncpoly>
<ncpoly>d*b-1</ncpoly>
<ncpoly>a*a*d*d*d*d-1</ncpoly>
<ncpoly>(a*b*a*b*a*b*b*b)^2-1</ncpoly>
<Comment>relation:(ababab^{3})^{2} = 1</Comment>
</basis>
<Comment>The partial LLex Gb has 96 elements</Comment>
<Comment>Other_groups5</Comment>
</FREEALGEBRA>
```

#### 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],[1]]);
Append(Relations,[[c,a],[1]]);
Append(Relations,[[b,d],[1]]);
Append(Relations,[[d,b],[1]]);

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

// 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],[1]]);

Return Relations;
EndDefine;

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

#### H in Symbolic Data Format

``` <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier">
<vars>a,b,c,d</vars>
<uptoDeg>13</uptoDeg>
<basis>
<ncpoly>a*c-1</ncpoly>
<ncpoly>c*a-1</ncpoly>
<ncpoly>b*d-1</ncpoly>
<ncpoly>d*b-1</ncpoly>
<Comment>polynomials to define inverse elements</Comment>
<ncpoly>a*a*b*b*b*b-1</ncpoly>
<ncpoly>(a*b*a*b*a*b*b*b)^2-1</ncpoly>
<Comment>relation:(ababab^{3})^{2}= 1</Comment>
</basis>
<Comment>The partial LLex Gb has 268 elements</Comment>
<Comment>Other_groups6</Comment>
</FREEALGEBRA>
```