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Properties

Label 12T11
Degree $12$
Order $24$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3 \times C_4$

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Show commands: Magma

magma: G := TransitiveGroup(12, 11);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $11$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3 \times C_4$
CHM label:   $S(3)[x]C(4)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,5)(2,10)(4,8)(7,11), (1,5,9)(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, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $D_{6}$

Low degree siblings

12T11, 24T12

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrder IndexRepresentative
1A $1^{12}$ $1$ $1$ $0$ $()$
2A $2^{6}$ $1$ $2$ $6$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
2B $2^{6}$ $3$ $2$ $6$ $( 1,11)( 2, 4)( 3, 9)( 5, 7)( 6,12)( 8,10)$
2C $2^{4},1^{4}$ $3$ $2$ $4$ $( 2, 6)( 3,11)( 5, 9)( 8,12)$
3A $3^{4}$ $2$ $3$ $8$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
4A1 $4^{3}$ $1$ $4$ $9$ $( 1,10, 7, 4)( 2,11, 8, 5)( 3,12, 9, 6)$
4A-1 $4^{3}$ $1$ $4$ $9$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$
4B1 $4^{3}$ $3$ $4$ $9$ $( 1,12, 7, 6)( 2, 5, 8,11)( 3,10, 9, 4)$
4B-1 $4^{3}$ $3$ $4$ $9$ $( 1,10, 7, 4)( 2, 3, 8, 9)( 5, 6,11,12)$
6A $6^{2}$ $2$ $6$ $10$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
12A1 $12$ $2$ $12$ $11$ $( 1, 8, 3,10, 5,12, 7, 2, 9, 4,11, 6)$
12A-1 $12$ $2$ $12$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$

magma: ConjugacyClasses(G);
 

Malle's constant $a(G)$:     $1/4$

Group invariants

Order:  $24=2^{3} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  24.5
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 6A 12A1 12A-1
Size 1 1 3 3 2 1 1 3 3 2 2 2
2 P 1A 1A 1A 1A 3A 2A 2A 2A 2A 3A 6A 6A
3 P 1A 2A 2B 2C 1A 4A-1 4A1 4B-1 4B1 2A 4A1 4A-1
Type
24.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1
24.5.1b R 1 1 −1 −1 1 −1 −1 1 1 1 −1 −1
24.5.1c R 1 1 −1 −1 1 1 1 −1 −1 1 1 1
24.5.1d R 1 1 1 1 1 −1 −1 −1 −1 1 −1 −1
24.5.1e1 C 1 −1 −1 1 1 −i i i −i −1 i −i
24.5.1e2 C 1 −1 −1 1 1 i −i −i i −1 −i i
24.5.1f1 C 1 −1 1 −1 1 −i i −i i −1 i −i
24.5.1f2 C 1 −1 1 −1 1 i −i i −i −1 −i i
24.5.2a R 2 2 0 0 −1 2 2 0 0 −1 −1 −1
24.5.2b R 2 2 0 0 −1 −2 −2 0 0 −1 1 1
24.5.2c1 C 2 −2 0 0 −1 −2i 2i 0 0 1 −i i
24.5.2c2 C 2 −2 0 0 −1 2i −2i 0 0 1 i −i

magma: CharacterTable(G);