# ApCoCoA-1:Dicyclic groups

From ApCoCoAWiki

#### Description

The dicyclic groups are non-abelian groups with order 4n. For n = 2 the dicyclic group is isomporphic to the quarternion group Q. Note that every element of this groups can be written uniquely as a^k x^j for 0 < k < 2n and j = 0 or 1.

Dic(n) = <a,b | a^{2n} = 1, a^{n} = b^{2}, b^{-1}ab = a^{-1}>

#### Reference

Coxeter, H. S. M., "7.1 The Cyclic and Dicyclic groups", Regular Complex Polytopes, Cambridge University, 1974.

#### Computation

/*Use the ApCoCoA package ncpoly.*/ // Number of Dicyclic group (note that the order is 4N) MEMORY.N:=5; Use ZZ/(2)[a,b]; NC.SetOrdering("LLEX"); Define CreateRelationsDicyclic() Relations:=[]; // Add the relation a^{2n} = 1 Append(Relations, [[a^(2*MEMORY.N)], [1]]); // Add the relation a^{n} = b^2 Append(Relations, [[a^(MEMORY.N)], [-b,b]]); // Add the relation b^{-1}ab = a^{-1} Append(Relations, [[b^(3),a,b],[a^(2*MEMORY.N-1)]]); Return Relations; EndDefine; Relations:=CreateRelationsDicyclic(); Relations; // Compute a Groebner basis Gb:=NC.GB(Relations); Gb; // Compute the values of the Hilbert-Dehn function NC.HF(Gb); // The order of the dicyclic group Sum(It);