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Properties

Label 12T7
Degree $12$
Order $24$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4 \times C_2$

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Show commands: Magma

magma: G := TransitiveGroup(12, 7);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_4 \times C_2$
CHM label:  $A_{4}(6)[x]2=[1/8.2^{6}]3$
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:  (2,8)(3,9)(4,10)(5,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: None

Degree 6: $C_6$, $A_4$, $A_4\times C_2$

Low degree siblings

6T6, 8T13, 12T6, 24T9

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

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:  24.13
magma: IdentifyGroup(G);
 
Character table:    
     2  3  3   1   1   1   1  3  3
     3  1  .   1   1   1   1  .  1

       1a 2a  6a  3a  6b  3b 2b 2c
    2P 1a 1a  3b  3b  3a  3a 1a 1a
    3P 1a 2a  2c  1a  2c  1a 2b 2c
    5P 1a 2a  6b  3b  6a  3a 2b 2c

X.1     1  1   1   1   1   1  1  1
X.2     1  1  -1   1  -1   1 -1 -1
X.3     1  1   A  -A  /A -/A -1 -1
X.4     1  1  /A -/A   A  -A -1 -1
X.5     1  1 -/A -/A  -A  -A  1  1
X.6     1  1  -A  -A -/A -/A  1  1
X.7     3 -1   .   .   .   .  1 -3
X.8     3 -1   .   .   .   . -1  3

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3

magma: CharacterTable(G);