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Properties

Label 12T119
Degree $12$
Order $216$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2:C_4\times S_3$

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Show commands: Magma

magma: G := TransitiveGroup(12, 119);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $119$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^2:C_4\times S_3$
CHM label:   $[3^{3}:2]4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,5)(2,10)(4,8)(7,11), (2,6,10)(3,7,11)(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)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$24$:  $S_3 \times C_4$
$36$:  $C_3^2:C_4$
$72$:  12T40

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: None

Low degree siblings

12T119, 18T95 x 2, 24T559 x 2, 27T85, 36T261 x 2, 36T262 x 2, 36T263 x 2, 36T295 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $27$ $2$ $( 4, 8)( 5, 9)( 6,10)( 7,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 3, 7,11)( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $8$ $3$ $( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 4, 4, 4 $ $27$ $4$ $( 1, 2, 3, 4)( 5, 6, 7, 8)( 9,10,11,12)$
$ 12 $ $18$ $12$ $( 1, 2, 3, 4, 5,10, 7,12, 9, 6,11, 8)$
$ 4, 4, 4 $ $9$ $4$ $( 1, 2, 3,12)( 4, 9, 6,11)( 5,10, 7, 8)$
$ 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)$
$ 6, 2, 2, 2 $ $12$ $6$ $( 1, 3)( 2, 4, 6,12,10, 8)( 5,11)( 7, 9)$
$ 6, 6 $ $18$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4,10,12, 6, 8)$
$ 6, 6 $ $12$ $6$ $( 1, 3, 9, 7, 5,11)( 2, 4,10, 8, 6,12)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 3)( 2,12)( 4,10)( 5,11)( 6, 8)( 7, 9)$
$ 4, 4, 4 $ $27$ $4$ $( 1, 4, 3, 2)( 5, 8, 7, 6)( 9,12,11,10)$
$ 12 $ $18$ $12$ $( 1, 4, 7,10, 5,12,11, 6, 9, 8, 3, 2)$
$ 4, 4, 4 $ $9$ $4$ $( 1, 4,11, 2)( 3,10, 5,12)( 6, 9, 8, 7)$
$ 3, 3, 3, 3 $ $4$ $3$ $( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4,12, 8)$
$ 3, 3, 3, 3 $ $2$ $3$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $216=2^{3} \cdot 3^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  216.156
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 3D 3E 4A1 4A-1 4B1 4B-1 6A 6B 6C 12A1 12A-1
Size 1 3 9 27 2 4 4 8 8 9 9 27 27 12 12 18 18 18
2 P 1A 1A 1A 1A 3A 3B 3C 3D 3E 2B 2B 2B 2B 3B 3C 3A 6C 6C
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 4A-1 4A1 4B-1 4B1 2A 2A 2B 4A1 4A-1
Type
216.156.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
216.156.1b R 1 −1 1 −1 1 1 1 1 1 −1 −1 1 1 −1 −1 1 −1 −1
216.156.1c R 1 −1 1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1
216.156.1d R 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1
216.156.1e1 C 1 −1 −1 1 1 1 1 1 1 −i i −i i −1 −1 −1 i −i
216.156.1e2 C 1 −1 −1 1 1 1 1 1 1 i −i i −i −1 −1 −1 −i i
216.156.1f1 C 1 1 −1 −1 1 1 1 1 1 −i i i −i 1 1 −1 i −i
216.156.1f2 C 1 1 −1 −1 1 1 1 1 1 i −i −i i 1 1 −1 −i i
216.156.2a R 2 0 2 0 −1 2 2 −1 −1 2 2 0 0 0 0 −1 −1 −1
216.156.2b R 2 0 2 0 −1 2 2 −1 −1 −2 −2 0 0 0 0 −1 1 1
216.156.2c1 C 2 0 −2 0 −1 2 2 −1 −1 −2i 2i 0 0 0 0 1 −i i
216.156.2c2 C 2 0 −2 0 −1 2 2 −1 −1 2i −2i 0 0 0 0 1 i −i
216.156.4a R 4 4 0 0 4 −2 1 1 −2 0 0 0 0 −2 1 0 0 0
216.156.4b R 4 4 0 0 4 1 −2 −2 1 0 0 0 0 1 −2 0 0 0
216.156.4c R 4 −4 0 0 4 −2 1 1 −2 0 0 0 0 2 −1 0 0 0
216.156.4d R 4 −4 0 0 4 1 −2 −2 1 0 0 0 0 −1 2 0 0 0
216.156.8a R 8 0 0 0 −4 −4 2 −1 2 0 0 0 0 0 0 0 0 0
216.156.8b R 8 0 0 0 −4 2 −4 2 −1 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);