# Properties

 Label 8T4 Degree $8$ Order $8$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $D_4$

# Learn more

Show commands: Magma

magma: G := TransitiveGroup(8, 4);

## Group action invariants

 Degree $n$: $8$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $4$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $D_4$ CHM label: $D_{8}(8)=[4]2$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $8$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,2,3,8)(4,5,6,7), (1,6)(2,5)(3,4)(7,8) magma: Generators(G);

## 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: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

## Low degree siblings

4T3 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Label Cycle Type Size Order Representative $1^{8}$ $1$ $1$ $()$ $4^{2}$ $2$ $4$ $(1,2,3,8)(4,5,6,7)$ $2^{4}$ $1$ $2$ $(1,3)(2,8)(4,6)(5,7)$ $2^{4}$ $2$ $2$ $(1,4)(2,7)(3,6)(5,8)$ $2^{4}$ $2$ $2$ $(1,5)(2,4)(3,7)(6,8)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $8=2^{3}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: $2$ Label: 8.3 magma: IdentifyGroup(G); Character table:

 1A 2A 2B 2C 4A Size 1 1 2 2 2 2 P 1A 1A 1A 1A 2A Type 8.3.1a R $1$ $1$ $1$ $1$ $1$ 8.3.1b R $1$ $1$ $−1$ $−1$ $1$ 8.3.1c R $1$ $1$ $−1$ $1$ $−1$ 8.3.1d R $1$ $1$ $1$ $−1$ $−1$ 8.3.2a R $2$ $−2$ $0$ $0$ $0$

magma: CharacterTable(G);