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Properties

Label 12T30
Degree $12$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(12, 30);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $30$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_4:C_4$
CHM label:  $1/2[1/4.4^{3}]S(3)$
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:  (3,9)(6,12), (1,8,7,2)(3,6,9,12)(4,11,10,5), (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)
$2$:  $C_2$
$4$:  $C_4$
$6$:  $S_3$
$12$:  $C_3 : C_4$
$24$:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_3$

Low degree siblings

12T27, 16T62, 24T51, 24T57

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $48=2^{4} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  48.30
magma: IdentifyGroup(G);
 
Character table:    
      2  4  4  4  3  3  3  3  1  1  4
      3  1  .  .  .  .  .  .  1  1  1

        1a 2a 2b 4a 4b 4c 4d 6a 3a 2c
     2P 1a 1a 1a 2a 2a 2c 2c 3a 3a 1a
     3P 1a 2a 2b 4b 4a 4d 4c 2c 1a 2c
     5P 1a 2a 2b 4a 4b 4c 4d 6a 3a 2c

X.1      1  1  1  1  1  1  1  1  1  1
X.2      1  1  1 -1 -1 -1 -1  1  1  1
X.3      1 -1  1  A -A -A  A -1  1 -1
X.4      1 -1  1 -A  A  A -A -1  1 -1
X.5      2 -2  2  .  .  .  .  1 -1 -2
X.6      2  2  2  .  .  .  . -1 -1  2
X.7      3 -1 -1 -1 -1  1  1  .  .  3
X.8      3 -1 -1  1  1 -1 -1  .  .  3
X.9      3  1 -1  A -A  A -A  .  . -3
X.10     3  1 -1 -A  A -A  A  .  . -3

A = -E(4)
  = -Sqrt(-1) = -i

magma: CharacterTable(G);