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Properties

Label 12T20
Degree $12$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times A_4$

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Show commands: Magma

magma: G := TransitiveGroup(12, 20);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $20$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times A_4$
CHM label:   $A(4)[x]C(3)$
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:  (1,7,10)(2,5,11)(3,6,9), (2,8,11)(3,6,12)(4,7,10), (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)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$12$:  $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: $A_4$

Degree 6: None

Low degree siblings

12T20 x 2, 18T8, 36T12

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

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:  36.11
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 6A1 6A-1
Size 1 3 1 1 4 4 4 4 4 4 3 3
2 P 1A 1A 3A-1 3A1 3C-1 3B-1 3D-1 3C1 3D1 3B1 3A1 3A-1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A
Type
36.11.1a R 1 1 1 1 1 1 1 1 1 1 1 1
36.11.1b1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1
36.11.1b2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3
36.11.1c1 C 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1
36.11.1c2 C 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
36.11.1d1 C 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
36.11.1d2 C 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
36.11.1e1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1
36.11.1e2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1
36.11.3a R 3 −1 3 3 0 0 0 0 0 0 −1 −1
36.11.3b1 C 3 −1 3ζ3−1 3ζ3 0 0 0 0 0 0 −ζ3 −ζ3−1
36.11.3b2 C 3 −1 3ζ3 3ζ3−1 0 0 0 0 0 0 −ζ3−1 −ζ3

magma: CharacterTable(G);