Linear representation theory of groups of order 16

Full listing
The following sets of degrees of irreducible representations works over any splitting field not of characteristic two.

See also nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order. This says that for groups of order $$p^k, 0 \le k \le 4$$, the nilpotency class of the group, and the order, together determine the degrees of irreducible representations. In particular, for groups of order 16, there are three cases: the abelian case, where there are 16 of degree one, the class two case, where there are 8 of degree one and 2 of degree two, and the class three case, where there are 4 of degree one and 3 of degree two.

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 16 mod 3, and hence congruent to 1 mod 3.

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, though this phenomenon does not occur for order 16.

For the first two Hall-Senior families $$\Gamma_1,\Gamma_2$$, 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 16 have $$4 \times (16/8) = 8$$ irreps of degree 1 and $$1 \times (16/8) = 2$$ irreps of degree 2.

For more background on the Hall-Senior families business, see Groups of order 16.

Grouping by degrees of irreducible representations
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.

For order 16 (and more generally for fixed order $$p^k$$ with $$p$$ prime and $$0 \le k \le 4$$), the degrees of irreducible representations uniquely determine the Hall-Senior family, i.e., different Hall-Senior families have different degrees of irreducible representations. This breaks down for $$p^5$$ and in particular for 32, where there are some cases of multiple Hall-Senior families having the same degrees of irreducible representations.

Characteristic zero case
Note that for the abelian cases, the smallest field of realization of representations is the same as the field generated by the character values, because the irreducible representations are all one-dimensional and can be identified with their character values.

Key shorthands in this table:


 * $$\mathbb{Q}(i)$$ stands for $$\mathbb{Q}(\sqrt{-1}) = \mathbb{Q}[t]/(t^2 + 1)$$
 * $$\mathbb{Q}(i,\sqrt{2})$$ stands for $$\mathbb{Q}(\sqrt{-1},\sqrt{-2}) = \mathbb{Q}(e^{\pi i/4})$$
 * $$\mathbb{Q}(\sqrt{2})$$ is the same as $$\mathbb{Q}[t]/(t^2 - 2)$$
 * $$\mathbb{Q}(\sqrt{-2})$$ is the same as $$\mathbb{Q}(\sqrt{2}i) = \mathbb{Q}[t]/(t^2 + 2)$$

Grouping by minimal splitting field
Note that since minimal splitting field need not be unique, some groups have multiple minimal splitting fields. All the minimal splitting fields for direct product of Q8 and Z2 and generalized quaternion group:Q16 are not mentioned.

Rationals and reals: properties
This table can be completely reconstructed based on the above/previous tables, but is included for additional clarity.