This is a coded (not in-built) function in GAP to group together all groups of a given order based on their degrees of irreducible representations.
The function is invoked as:
where is a natural number. The output is a list of pairs of lists, where each pair is as follows:
- The first member of the pair is itself a list of pairs describing a collection of degrees of irreducible representations over : the first member of each pair is a degree and the second member is the number of irreducible representations of that degree.
- The second member of the pair is the list of second parts of GAP IDs of groups of order having those degrees of irreducible representations.
IrrepDegreeGroupingFull := function(n) local L,M,m,N; L := List(AllSmallGroups(n),CharacterDegrees); M := SortedList(Set(L)); m := Length(L); N := List(M,i -> [i,Filtered([1..m],j->L[j]=i)]); return(SortedList(N)); end;;
Examples of usage
|1||IrrepDegreeGroupingFull(1)||[ [ [ [ 1, 1 ] ], [ 1 ] ] ]||The group with GAP ID has 1 irreducible representation of degree 1.|
|8||IrrepDegreeGroupingFull(8)||[ [ [ [ 1, 4 ], [ 2, 1 ] ], [ 3, 4 ] ], [ [ [ 1, 8 ] ], [ 1, 2, 5 ] ] ]||The groups with GAP IDs and both have four irreducible representations of degree 1 and one irreducible representation of degree 2. The groups with GAP IDs , , and each have 8 irreducible representations of degree 1.|