↑

Properties

Label 12T131
Degree $12$
Order $324$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\wr C_4$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(12, 131);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $131$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\wr C_4$
CHM label:   $[3^{4}]4=3wr4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $S_3$, $C_6$
$12$:  $C_{12}$, $C_3 : C_4$
$18$:  $S_3\times C_3$
$36$:  $C_3^2:C_4$, $C_3\times (C_3 : C_4)$
$108$:  12T72, 12T73

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: None

Low degree siblings

12T131 x 3, 18T123 x 4, 36T497 x 4, 36T514 x 2, 36T527 x 2, 36T532 x 4, 36T536 x 4, 36T543 x 4, 36T544 x 8

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 $ $1$ $1$ $()$
$ 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 4, 8,12)$
$ 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 3, 7,11)( 4, 8,12)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 3, 7,11)( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 3,11, 7)( 4, 8,12)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 3,11, 7)( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $2$ $3$ $( 2, 6,10)( 4, 8,12)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 3, 7,11)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 3,11, 7)( 4, 8,12)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 3,11, 7)( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $2$ $3$ $( 2,10, 6)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2,10, 6)( 3, 7,11)( 4, 8,12)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2,10, 6)( 3, 7,11)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2,10, 6)( 3,11, 7)( 4, 8,12)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$ 4, 4, 4 $ $27$ $4$ $( 1, 2, 3, 4)( 5, 6, 7, 8)( 9,10,11,12)$
$ 12 $ $27$ $12$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$
$ 12 $ $27$ $12$ $( 1, 2, 3, 4, 9,10,11,12, 5, 6, 7, 8)$
$ 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)$
$ 6, 2, 2, 2 $ $18$ $6$ $( 1, 3)( 2, 4, 6, 8,10,12)( 5, 7)( 9,11)$
$ 6, 2, 2, 2 $ $18$ $6$ $( 1, 3)( 2, 4,10,12, 6, 8)( 5, 7)( 9,11)$
$ 6, 6 $ $9$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
$ 6, 6 $ $18$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4,10,12, 6, 8)$
$ 6, 6 $ $9$ $6$ $( 1, 3, 9,11, 5, 7)( 2, 4,10,12, 6, 8)$
$ 4, 4, 4 $ $27$ $4$ $( 1, 4, 3, 2)( 5, 8, 7, 6)( 9,12,11,10)$
$ 12 $ $27$ $12$ $( 1, 4, 7, 6, 5, 8,11,10, 9,12, 3, 2)$
$ 12 $ $27$ $12$ $( 1, 4,11,10, 9,12, 7, 6, 5, 8, 3, 2)$
$ 3, 3, 3, 3 $ $1$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 3, 3, 3, 3 $ $4$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4,12, 8)$
$ 3, 3, 3, 3 $ $4$ $3$ $( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4,12, 8)$
$ 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$
$ 3, 3, 3, 3 $ $4$ $3$ $( 1, 5, 9)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$ 3, 3, 3, 3 $ $1$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $324=2^{2} \cdot 3^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  324.162
magma: IdentifyGroup(G);
 
Character table:    36 x 36 character table

magma: CharacterTable(G);