# Difference between revisions of "ApCoCoA-1:Other1 groups"

### Other Groups

#### Description

This group has the following representation:

``` G = <a,b | a^{2}b^{-6} = (ab^{-1})^{3}ab^{-2}ab^{k}a^{-1}b = 1>
```

where k is congruent to 3 mod 6.

#### Reference

No reference available

#### Computation

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

//K is congruent to 3 mod 6
MEMORY.K:=3;
// a is invers to c and b is invers to d
Use ZZ/(2)[a,b,c,d];
NC.SetOrdering("LLEX");
Define CreateRelationsOther1()
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],]);

Append(Relations,[[a,a,d,d,d,d,d,d],]);

// add the relation (ab^{-1})^{3}ab^{-2}ab^{k}a^{-1}b = 1
Append(Relations,[[a,d,a,d,a,d,a,d,d,a,b^MEMORY.K,c,b],]);

Return Relations;
EndDefine;

Relations:=CreateRelationsOther1();
Gb:=NC.GB(Relations,31,1,100,1000);
```

#### Examples in Symbolic Data Format

##### Other group 1 k=3
``` <FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier">
<vars>a,b,c,d</vars>
<uptoDeg>10</uptoDeg>
<basis>
<ncpoly>a*c-1</ncpoly>
<ncpoly>c*a-1</ncpoly>
<ncpoly>b*d-1</ncpoly>
<ncpoly>d*b-1</ncpoly>
<ncpoly>(a^2)*(d^6)-1</ncpoly>
<ncpoly>((a*d)^3)*a*d*d*a*(b^3)*c*b-1</ncpoly>
</basis>
<Comment>Other_groups1k3</Comment>
</FREEALGEBRA>
```
##### Other group 1 k=39
``` <FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier">
<vars>a,b,c,d</vars>
<uptoDeg>10</uptoDeg>
<basis>
<ncpoly>a*c-1</ncpoly>
<ncpoly>c*a-1</ncpoly>
<ncpoly>b*d-1</ncpoly>
<ncpoly>d*b-1</ncpoly>
<ncpoly>(a^2)*(d^6)-1</ncpoly>
<ncpoly>((a*d)^3)*a*d*d*a*(b^39)*c*b-1</ncpoly>
</basis>
<Comment>The partial LLex Gb has 234 elements</Comment>
<Comment>Other_groups1k39</Comment>
</FREEALGEBRA>
```
##### Other group 1 k=63
``` <FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier">
<vars>a,b,c,d</vars>
<uptoDeg>9</uptoDeg>
<basis>
<ncpoly>a*c-1</ncpoly>
<ncpoly>c*a-1</ncpoly>
<ncpoly>b*d-1</ncpoly>
<ncpoly>d*b-1</ncpoly>
<ncpoly>(a^2)*(d^6)-1</ncpoly>
<ncpoly>((a*d)^3)*a*d*d*a*(b^63)*c*b-1</ncpoly>
</basis>
<Comment>The partial LLex Gb has 137 elements</Comment>
<Comment>Other_groups1k63</Comment>
</FREEALGEBRA>
```