Properties

Label 12T19
Degree $12$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times (C_3 : C_4)$

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Show commands: Magma

magma: G := TransitiveGroup(12, 19);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $19$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times (C_3 : C_4)$
CHM label:   $[3^{2}]4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (2,6,10)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: $S_3\times C_3$

Low degree siblings

36T5

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{12}$ $1$ $1$ $0$ $()$
2A $2^{6}$ $1$ $2$ $6$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
3A1 $3^{4}$ $1$ $3$ $8$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
3A-1 $3^{4}$ $1$ $3$ $8$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
3B $3^{2},1^{6}$ $2$ $3$ $4$ $( 1, 9, 5)( 3,11, 7)$
3C1 $3^{4}$ $2$ $3$ $8$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$
3C-1 $3^{2},1^{6}$ $2$ $3$ $4$ $( 2, 6,10)( 4, 8,12)$
4A1 $4^{3}$ $3$ $4$ $9$ $( 1,10, 7, 4)( 2,11, 8, 5)( 3,12, 9, 6)$
4A-1 $4^{3}$ $3$ $4$ $9$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$
6A1 $6^{2}$ $1$ $6$ $10$ $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$
6A-1 $6^{2}$ $1$ $6$ $10$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
6B $6^{2}$ $2$ $6$ $10$ $( 1,11, 9, 7, 5, 3)( 2, 4, 6, 8,10,12)$
6C1 $6,2^{3}$ $2$ $6$ $8$ $( 1, 3, 5, 7, 9,11)( 2, 8)( 4,10)( 6,12)$
6C-1 $6,2^{3}$ $2$ $6$ $8$ $( 1, 7)( 2,12,10, 8, 6, 4)( 3, 9)( 5,11)$
12A1 $12$ $3$ $12$ $11$ $( 1,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
12A-1 $12$ $3$ $12$ $11$ $( 1, 6,11, 4, 9, 2, 7,12, 5,10, 3, 8)$
12A5 $12$ $3$ $12$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$
12A-5 $12$ $3$ $12$ $11$ $( 1, 8, 3,10, 5,12, 7, 2, 9, 4,11, 6)$

Malle's constant $a(G)$:     $1/4$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  36.6
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B 3C1 3C-1 4A1 4A-1 6A1 6A-1 6B 6C1 6C-1 12A1 12A-1 12A5 12A-5
Size 1 1 1 1 2 2 2 3 3 1 1 2 2 2 3 3 3 3
2 P 1A 1A 3A-1 3A1 3C1 3B 3C-1 2A 2A 3A1 3A-1 3B 3C1 3C-1 6A1 6A1 6A-1 6A-1
3 P 1A 2A 1A 1A 1A 1A 1A 4A-1 4A1 2A 2A 2A 2A 2A 4A1 4A-1 4A-1 4A1
Type
36.6.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
36.6.1b R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
36.6.1c1 C 1 1 ζ31 ζ3 1 ζ31 ζ3 1 1 ζ3 ζ31 1 ζ3 ζ31 ζ31 ζ3 ζ3 ζ31
36.6.1c2 C 1 1 ζ3 ζ31 1 ζ3 ζ31 1 1 ζ31 ζ3 1 ζ31 ζ3 ζ3 ζ31 ζ31 ζ3
36.6.1d1 C 1 1 1 1 1 1 1 i i 1 1 1 1 1 i i i i
36.6.1d2 C 1 1 1 1 1 1 1 i i 1 1 1 1 1 i i i i
36.6.1e1 C 1 1 ζ31 ζ3 1 ζ31 ζ3 1 1 ζ3 ζ31 1 ζ3 ζ31 ζ31 ζ3 ζ3 ζ31
36.6.1e2 C 1 1 ζ3 ζ31 1 ζ3 ζ31 1 1 ζ31 ζ3 1 ζ31 ζ3 ζ3 ζ31 ζ31 ζ3
36.6.1f1 C 1 1 ζ122 ζ124 1 ζ122 ζ124 ζ123 ζ123 ζ124 ζ122 1 ζ124 ζ122 ζ125 ζ12 ζ12 ζ125
36.6.1f2 C 1 1 ζ124 ζ122 1 ζ124 ζ122 ζ123 ζ123 ζ122 ζ124 1 ζ122 ζ124 ζ12 ζ125 ζ125 ζ12
36.6.1f3 C 1 1 ζ122 ζ124 1 ζ122 ζ124 ζ123 ζ123 ζ124 ζ122 1 ζ124 ζ122 ζ125 ζ12 ζ12 ζ125
36.6.1f4 C 1 1 ζ124 ζ122 1 ζ124 ζ122 ζ123 ζ123 ζ122 ζ124 1 ζ122 ζ124 ζ12 ζ125 ζ125 ζ12
36.6.2a R 2 2 2 2 1 1 1 0 0 2 2 1 1 1 0 0 0 0
36.6.2b S 2 2 2 2 1 1 1 0 0 2 2 1 1 1 0 0 0 0
36.6.2c1 C 2 2 2ζ31 2ζ3 1 ζ31 ζ3 0 0 2ζ3 2ζ31 1 ζ3 ζ31 0 0 0 0
36.6.2c2 C 2 2 2ζ3 2ζ31 1 ζ3 ζ31 0 0 2ζ31 2ζ3 1 ζ31 ζ3 0 0 0 0
36.6.2d1 C 2 2 2ζ31 2ζ3 1 ζ31 ζ3 0 0 2ζ3 2ζ31 1 ζ3 ζ31 0 0 0 0
36.6.2d2 C 2 2 2ζ3 2ζ31 1 ζ3 ζ31 0 0 2ζ31 2ζ3 1 ζ31 ζ3 0 0 0 0

magma: CharacterTable(G);