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Properties

Label 12T9
Degree $12$
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(12, 9);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_4$
CHM label:  $1/2[1/8.2^{6}]S(3)=S_{4}(12e)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2)(3,12)(4,11)(5,10)(6,9)(7,8), (1,7)(3,9)(4,10)(6,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
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: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_3$, $S_4$, $S_4$

Low degree siblings

4T5, 6T7, 6T8, 8T14, 12T8, 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, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 8)( 3, 9)( 5,11)( 6,12)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 2, 7, 8)( 3, 6)( 4,11,10, 5)( 9,12)$
$ 3, 3, 3, 3 $ $8$ $3$ $( 1, 3,11)( 2, 4, 6)( 5, 7, 9)( 8,10,12)$

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  3  2  2  .
     3  1  .  .  .  1

       1a 2a 2b 4a 3a
    2P 1a 1a 1a 2a 3a
    3P 1a 2a 2b 4a 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);