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>

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 CreateRelationsOther4()
   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:=CreateRelationsOther4();
 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 CreateRelationsOther4()
   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:=CreateRelationsOther4();
 GB:=NC.GB(Relations,31,1,100,1000);