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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);
 
Nilpotency class:  $2$
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

|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

Cycle TypeSizeOrderRepresentative
$ 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 $ $1$ $2$ $(1,3)(2,4)$

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);
 
Label:  8.3
magma: IdentifyGroup(G);
 
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

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.