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Properties

Label 6T11
6T11 1 3 1->3 5 1->5 2 4 2->4 2->4 3->5 6 3->6 4->6 5->1 6->2
Degree $6$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_4\times C_2$

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Copy content magma:G := TransitiveGroup(6, 11);
 

Group invariants

Abstract group:  $S_4\times C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $48=2^{4} \cdot 3$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $6$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $11$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $2S_{4}(6) = [2^{3}]S(3) = 2 wr S(3)$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,5)(2,4)$, $(1,3,5)(2,4,6)$, $(3,6)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$24$:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Low degree siblings

6T11, 8T24 x 2, 12T21, 12T22, 12T23 x 2, 12T24 x 2, 16T61, 24T46, 24T47, 24T48 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{6}$ $1$ $1$ $0$ $()$
2A $2^{3}$ $1$ $2$ $3$ $(1,4)(2,5)(3,6)$
2B $2,1^{4}$ $3$ $2$ $1$ $(2,5)$
2C $2^{2},1^{2}$ $3$ $2$ $2$ $(1,4)(2,5)$
2D $2^{3}$ $6$ $2$ $3$ $(1,2)(3,6)(4,5)$
2E $2^{2},1^{2}$ $6$ $2$ $2$ $(1,2)(4,5)$
3A $3^{2}$ $8$ $3$ $4$ $(1,2,6)(3,4,5)$
4A $4,2$ $6$ $4$ $4$ $(1,2,4,5)(3,6)$
4B $4,1^{2}$ $6$ $4$ $3$ $(1,2,4,5)$
6A $6$ $8$ $6$ $5$ $(1,3,2,4,6,5)$

Malle's constant $a(G)$:     $1$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 3A 4A 4B 6A
Size 1 1 3 3 6 6 8 6 6 8
2 P 1A 1A 1A 1A 1A 1A 3A 2C 2C 3A
3 P 1A 2A 2B 2C 2D 2E 1A 4A 4B 2A
Type
48.48.1a R 1 1 1 1 1 1 1 1 1 1
48.48.1b R 1 −1 −1 1 −1 1 1 1 −1 −1
48.48.1c R 1 −1 −1 1 1 −1 1 −1 1 −1
48.48.1d R 1 1 1 1 −1 −1 1 −1 −1 1
48.48.2a R 2 2 2 2 0 0 −1 0 0 −1
48.48.2b R 2 −2 −2 2 0 0 −1 0 0 1
48.48.3a R 3 3 −1 −1 1 1 0 −1 −1 0
48.48.3b R 3 3 −1 −1 −1 −1 0 1 1 0
48.48.3c R 3 −3 1 −1 −1 1 0 −1 1 0
48.48.3d R 3 −3 1 −1 1 −1 0 1 −1 0

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{6} + x^{2} + t$ Copy content Toggle raw display