Group action invariants
Degree $n$: | $4$ | |
Transitive number $t$: | $3$ | |
Group: | $D_{4}$ | |
CHM label: | $D(4)$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $2$ | |
$\card{\Aut(F/K)}$: | $2$ | |
Generators: | (1,2,3,4), (1,3) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Low degree siblings
4T3, 8T4Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 1, 1 $ | $2$ | $2$ | $(2,4)$ |
$ 2, 2 $ | $2$ | $2$ | $(1,2)(3,4)$ |
$ 4 $ | $2$ | $4$ | $(1,2,3,4)$ |
$ 2, 2 $ | $1$ | $2$ | $(1,3)(2,4)$ |
Group invariants
Order: | $8=2^{3}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
Label: | 8.3 |
Character table: |
2 3 2 2 2 3 1a 2a 2b 4a 2c 2P 1a 1a 1a 2c 1a 3P 1a 2a 2b 4a 2c X.1 1 1 1 1 1 X.2 1 -1 -1 1 1 X.3 1 -1 1 -1 1 X.4 1 1 -1 -1 1 X.5 2 . . . -2 |
Additional information
If a degree four extension of fields has this, 4T3, as its Galois group, then there is a non-isomorphic degree four extension with the same normal closure. 4T3 give the lowest degree example of this phenomenon.