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Properties

Label 6T8
Degree $6$
Order $24$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_4$

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Show commands: Magma

magma: G := TransitiveGroup(6, 8);
 

Group action invariants

Degree $n$:  $6$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $8$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_4$
CHM label:  $S_{4}(6c) = 1/2[2^{3}]S(3)$
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:  (1,5)(2,4)(3,6), (1,4)(2,5), (1,3,5)(2,4,6)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Low degree siblings

4T5, 6T7, 8T14, 12T8, 12T9, 24T10

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 $ $1$ $1$ $()$
$ 4, 1, 1 $ $6$ $4$ $(2,3,5,6)$
$ 2, 2, 1, 1 $ $3$ $2$ $(2,5)(3,6)$
$ 2, 2, 2 $ $6$ $2$ $(1,2)(3,6)(4,5)$
$ 3, 3 $ $8$ $3$ $(1,2,3)(4,5,6)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $24=2^{3} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  24.12
magma: IdentifyGroup(G);
 
Character table:    
     2  3  2  3  2  .
     3  1  .  .  .  1

       1a 4a 2a 2b 3a
    2P 1a 2a 1a 1a 3a
    3P 1a 4a 2a 2b 1a

X.1     1  1  1  1  1
X.2     1 -1  1 -1  1
X.3     2  .  2  . -1
X.4     3 -1 -1  1  .
X.5     3  1 -1 -1  .

magma: CharacterTable(G);