Linear representation theory of groups of order 32

Full listing
Here are the degrees of irreducible representations for all groups of order 32:

Grouping by Hall-Senior families
Note that isoclinic groups have same proportions of degrees of irreducible representations, and in particular isoclinic groups of the same order have precisely the same degrees of irreducible representations. Thus, all groups in the same Hall-Senior family have the same degrees of irreducible representations. However, the same multiset of degrees of irreducible representations could be attained by more than one Hall-Senior family, as we see in the table below.

For the first three Hall-Senior families $$\Gamma_1,\Gamma_2,\Gamma_3$$, there are isoclinic groups of smaller order, hence the degrees of irreducible representations can be computed by first computing the degrees of irreducible representations of those isoclinic groups of smaller order and then scaling up the proportions based on the order. For instance, dihedral group:D8 of order 8 and family $$\Gamma_2$$ has 4 irreps of degree 1 and 1 of degree 2, so the groups in family $$\Gamma_2$$ and of order 32 have $$4 \times (32/8) = 16$$ irreps of degree 1 and $$1 \times (32/8) = 4$$ irreps of degree 2.

For more background on the Hall-Senior families business, see Groups of order 32.

Grouping by degrees of irreducible representations
Note that since number of conjugacy classes in group of prime power order is congruent to order of group modulo prime-square minus one, all the values for the total number of irreducible representations, which equals the number of conjugacy classes, is congruent to 32 mod 3, and hence congruent to 2 mod 3.

As noted above, isoclinic groups have same proportions of degrees of irreducible representations, and in particular isoclinic groups of the same order have precisely the same degrees of irreducible representations. Thus, all groups in the same Hall-Senior family have the same degrees of irreducible representations. However, the same multiset of degrees of irreducible representations could be attained by more than one Hall-Senior family, as we see in the table below.

For more background on the Hall-Senior families business, see Groups of order 32.

For more on the subgroup structure stuff (the two right-most columns) see subgroup structure of groups of order 32.

Here is the GAP code to generate this information:

We use the (not in-built, but coded) function IrrepDegreeGroupingFull (follow link to get code) to get the information on groups of order 32 grouped by their degrees of irreducible representations:

gap> IrrepDegreeGroupingFull(32); [ [ [ [ 1, 4 ], [ 2, 7 ] ], [ 18, 19, 20 ] ], [ [ [ 1, 8 ], [ 2, 2 ], [ 4, 1 ] ], [ 6, 7, 8, 43, 44 ] ],  [ [ [ 1, 8 ], [ 2, 6 ] ],      [ 9, 10, 11, 13, 14, 15, 27, 28, 29, 30, 31, 32, 33, 34, 35, 39, 40,          41, 42 ] ],  [ [ [ 1, 16 ], [ 2, 4 ] ], [ 2, 4, 5, 12, 17, 22, 23, 24, 25, 26, 37, 38,          46, 47, 48 ] ], [ [ [ 1, 16 ], [ 4, 1 ] ], [ 49, 50 ] ],  [ [ [ 1, 32 ] ], [ 1, 3, 16, 21, 36, 45, 51 ] ] ]

Correspondence between degrees of irreducible representations and conjugacy class sizes
See also element structure of groups of order 32.

For groups of order 32, 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