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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

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

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 ζ3−1 ζ3 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
36.6.1c2 C 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 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 ζ3−1 ζ3 1 ζ3−1 ζ3 −1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
36.6.1e2 C 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 −1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −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 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 0 0 0
36.6.2c2 C 2 2 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 0 0 0
36.6.2d1 C 2 −2 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 0 −2ζ3 −2ζ3−1 1 ζ3 ζ3−1 0 0 0 0
36.6.2d2 C 2 −2 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 0 −2ζ3−1 −2ζ3 1 ζ3−1 ζ3 0 0 0 0

magma: CharacterTable(G);