Properties

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

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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

LabelCycle TypeSizeOrderRepresentative
$1^{8}$ $1$ $1$ $()$
$4^{2}$ $2$ $4$ $(1,2,3,8)(4,5,6,7)$
$2^{4}$ $1$ $2$ $(1,3)(2,8)(4,6)(5,7)$
$2^{4}$ $2$ $2$ $(1,4)(2,7)(3,6)(5,8)$
$2^{4}$ $2$ $2$ $(1,5)(2,4)(3,7)(6,8)$

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);