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Properties

Label 12T150
Degree $12$
Order $384$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_4\wr S_3$

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Show commands: Magma

magma: G := TransitiveGroup(12, 150);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $150$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4\wr S_3$
CHM label:   $[4^{3}]S(3)=4wrS(3)$
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:  (3,6,9,12), (1,5)(2,10)(4,8)(7,11), (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)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$24$:  $S_4$, $S_3 \times C_4$
$48$:  $S_4\times C_2$
$96$:  12T53, 12T62
$192$:  12T95

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4\times C_2$

Low degree siblings

12T150 x 3, 24T835, 24T857, 24T861, 24T1256 x 2, 24T1257 x 2, 24T1258 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $4$ $( 3, 6, 9,12)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 3, 9)( 6,12)$
$ 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $4$ $( 3,12, 9, 6)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $12$ $2$ $( 2, 3)( 5, 6)( 8, 9)(11,12)$
$ 8, 1, 1, 1, 1 $ $12$ $8$ $( 2, 3, 5, 6, 8, 9,11,12)$
$ 4, 4, 1, 1, 1, 1 $ $12$ $4$ $( 2, 3, 8, 9)( 5, 6,11,12)$
$ 8, 1, 1, 1, 1 $ $12$ $8$ $( 2, 3,11,12, 8, 9, 5, 6)$
$ 4, 4, 1, 1, 1, 1 $ $3$ $4$ $( 2, 5, 8,11)( 3, 6, 9,12)$
$ 4, 2, 2, 1, 1, 1, 1 $ $6$ $4$ $( 2, 5, 8,11)( 3, 9)( 6,12)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 2, 5, 8,11)( 3,12, 9, 6)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 8)( 3, 9)( 5,11)( 6,12)$
$ 4, 2, 2, 1, 1, 1, 1 $ $6$ $4$ $( 2, 8)( 3,12, 9, 6)( 5,11)$
$ 4, 4, 1, 1, 1, 1 $ $3$ $4$ $( 2,11, 8, 5)( 3,12, 9, 6)$
$ 4, 2, 2, 2, 2 $ $12$ $4$ $( 1, 2)( 3, 6, 9,12)( 4, 5)( 7, 8)(10,11)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3, 9)( 4, 5)( 6,12)( 7, 8)(10,11)$
$ 4, 2, 2, 2, 2 $ $12$ $4$ $( 1, 2)( 3,12, 9, 6)( 4, 5)( 7, 8)(10,11)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)$
$ 12 $ $32$ $12$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$
$ 6, 6 $ $32$ $6$ $( 1, 2, 3, 7, 8, 9)( 4, 5, 6,10,11,12)$
$ 12 $ $32$ $12$ $( 1, 2, 3,10,11,12, 7, 8, 9, 4, 5, 6)$
$ 8, 4 $ $12$ $8$ $( 1, 2, 4, 5, 7, 8,10,11)( 3, 6, 9,12)$
$ 8, 2, 2 $ $12$ $8$ $( 1, 2, 4, 5, 7, 8,10,11)( 3, 9)( 6,12)$
$ 8, 4 $ $12$ $8$ $( 1, 2, 4, 5, 7, 8,10,11)( 3,12, 9, 6)$
$ 4, 4, 4 $ $12$ $4$ $( 1, 2, 7, 8)( 3, 6, 9,12)( 4, 5,10,11)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 2, 7, 8)( 3, 9)( 4, 5,10,11)( 6,12)$
$ 4, 4, 4 $ $12$ $4$ $( 1, 2, 7, 8)( 3,12, 9, 6)( 4, 5,10,11)$
$ 8, 4 $ $12$ $8$ $( 1, 2,10,11, 7, 8, 4, 5)( 3, 6, 9,12)$
$ 8, 2, 2 $ $12$ $8$ $( 1, 2,10,11, 7, 8, 4, 5)( 3, 9)( 6,12)$
$ 8, 4 $ $12$ $8$ $( 1, 2,10,11, 7, 8, 4, 5)( 3,12, 9, 6)$
$ 4, 4, 4 $ $1$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$
$ 4, 4, 2, 2 $ $3$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 9)( 6,12)$
$ 4, 4, 4 $ $3$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3,12, 9, 6)$
$ 4, 2, 2, 2, 2 $ $3$ $4$ $( 1, 4, 7,10)( 2, 8)( 3, 9)( 5,11)( 6,12)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 4, 7,10)( 2, 8)( 3,12, 9, 6)( 5,11)$
$ 4, 4, 4 $ $3$ $4$ $( 1, 4, 7,10)( 2,11, 8, 5)( 3,12, 9, 6)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 4, 2, 2, 2, 2 $ $3$ $4$ $( 1, 7)( 2, 8)( 3,12, 9, 6)( 4,10)( 5,11)$
$ 4, 4, 2, 2 $ $3$ $4$ $( 1, 7)( 2,11, 8, 5)( 3,12, 9, 6)( 4,10)$
$ 4, 4, 4 $ $1$ $4$ $( 1,10, 7, 4)( 2,11, 8, 5)( 3,12, 9, 6)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $384=2^{7} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  384.5557
magma: IdentifyGroup(G);
 
Character table:    40 x 40 character table

magma: CharacterTable(G);