Properties

Label 4T3
Degree $4$
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(4, 3);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$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, 8T4

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrder IndexRepresentative
1A $1^{4}$ $1$ $1$ $0$ $()$
2A $2^{2}$ $1$ $2$ $2$ $(1,3)(2,4)$
2B $2^{2}$ $2$ $2$ $2$ $(1,2)(3,4)$
2C $2,1^{2}$ $2$ $2$ $1$ $(2,4)$
4A $4$ $2$ $4$ $3$ $(1,2,3,4)$

Malle's constant $a(G)$:     $1$

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);
 

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.