Linear representation theory of groups of order 8

Degrees of irreducible representations
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 8, there are only two cases: abelian, where all the irreducible representations have degree 1, and the class two groups, where there is one irreducible representation of degree 2.  

Grouping by minimal splitting field
Note that since minimal splitting field need not be unique, some groups have multiple minimal splitting fields. Among the groups of order 8, the only group with multiple minimal splitting fields is the quaternion group.

General case
Note that because sufficiently large implies splitting, the polynomial $$t^d - 1$$ splitting where $$d$$ is the exponent of the group is a sufficient condition for being a splitting field. However, it is not a necessary condition in general. For groups of order 8, the condition turns out to be necessary except in the case of dihedral group:D8 and quaternion group.

Here, we consider fields of characteristic not equal to $$2$$.

General case
Here, we consider rings of characteristic either zero or a prime power $$p^k$$, where $$p$$ is odd.

Orbits under automorphism group
We provide here the structure and sizes of orbits under the action of the automorphism group. The information presented below is valid in characteristic zero.

Action of Galois automorphisms
We provide here the structure and sizes of orbits under the action of Galois automorphisms. The information presented below is valid in characteristic zero.