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Properties

Label 12T31
Degree $12$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_4^2:C_3$

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Show commands: Magma

magma: G := TransitiveGroup(12, 31);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $31$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4^2:C_3$
CHM label:  $[4^{2}]3$
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:  (1,10,7,4)(3,6,9,12), (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)
$3$:  $C_3$
$12$:  $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4$

Low degree siblings

12T31, 16T63, 24T58

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 $ $1$ $1$ $()$
$ 4, 4, 1, 1, 1, 1 $ $3$ $4$ $( 2, 5, 8,11)( 3,12, 9, 6)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 8)( 3, 9)( 5,11)( 6,12)$
$ 4, 4, 1, 1, 1, 1 $ $3$ $4$ $( 2,11, 8, 5)( 3, 6, 9,12)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)$
$ 4, 4, 2, 2 $ $3$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 9)( 6,12)$
$ 4, 4, 2, 2 $ $3$ $4$ $( 1, 7)( 2,11, 8, 5)( 3,12, 9, 6)( 4,10)$

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:  48.3
magma: IdentifyGroup(G);
 
Character table:    
     2  4  4  4  4  .  .  4  4
     3  1  .  .  .  1  1  .  .

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

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

A = -1+2*E(4)
  = -1+2*Sqrt(-1) = -1+2i
B = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3

magma: CharacterTable(G);