Linear representation theory of groups of order 12

Full listing
Here is the GAP code to generate these data:

gap> List([1..Length(AllSmallGroups(12))],i -> [i,CharacterDegrees(SmallGroup(12,i))]); [ [ 1, [ [ 1, 4 ], [ 2, 2 ] ] ], [ 2, [ [ 1, 12 ] ] ], [ 3, [ [ 1, 3 ], [ 3, 1 ] ] ], [ 4, [ [ 1, 4 ], [ 2, 2 ] ] ], [ 5, [ [ 1, 12 ] ] ] ]

The code uses GAP's CharacterDegrees and SmallGroup functions.

Grouping by degrees of irreducible representations
Here is the GAP code to generate these data:

The code uses the coded (not in-built) function IrrepDegreeGroupingFull (follow link to get code) as follows:

gap> IrrepDegreeGroupingFull(12); [ [ [ [ 1, 3 ], [ 3, 1 ] ], [ 3 ] ], [ [ [ 1, 4 ], [ 2, 2 ] ], [ 1, 4 ] ], [ [ [ 1, 12 ] ], [ 2, 5 ] ] ]

Correspondence between degrees of irreducible representations and conjugacy class sizes
See also element structure of groups of order 12.

For groups of order 12, it is true that the list of conjugacy class sizes completely determines the list of degrees of irreducible representations, and vice versa. The details are given below. The middle column, which is the total number of each, separates the description of the list of conjugacy class sizes and the list of degrees of irreducible representations:

Note that the phenomenon of the conjugacy class size statistics and degrees of irreducible representations determining one another is not true for all orders:


 * Degrees of irreducible representations need not determine conjugacy class size statistics
 * Conjugacy class size statistics need not determine degrees of irreducible representations