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Properties

Label 4T3
4T3 1 2 1->2 3 1->3 2->3 4 3->4 4->1
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

Copy content magma:G := TransitiveGroup(4, 3);
 

Group invariants

Abstract group:  $D_{4}$
Copy content magma:IdentifyGroup(G);
 
Order:  $8=2^{3}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:  $2$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $4$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $3$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $D(4)$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,2,3,4)$, $(1,3)$
Copy content 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 TypeSizeOrderIndexRepresentative
1A $1^{4}$ $1$ $1$ $0$ $()$
2A $2^{2}$ $1$ $2$ $2$ $(1,3)(2,4)$
2B $2,1^{2}$ $2$ $2$ $1$ $(2,4)$
2C $2^{2}$ $2$ $2$ $2$ $(1,2)(3,4)$
4A $4$ $2$ $4$ $3$ $(1,2,3,4)$

Malle's constant $a(G)$:     $1$

Copy content magma:ConjugacyClasses(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

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{4} +s x^{2} +t$ Copy content Toggle raw display
The polynomial $f_{1}$ is generic for any base field $K$ of characteristic $\neq$ 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.