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Properties

Label 6T16
6T16 1 2 1->2 1->2 3 2->3 4 3->4 5 4->5 6 5->6 6->1
Degree $6$
Order $720$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $S_6$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(6, 16);
 

Group invariants

Abstract group:  $S_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $720=2^{4} \cdot 3^{2} \cdot 5$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
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 $t$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $S6$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  yes
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,2)$, $(1,2,3,4,5,6)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Low degree siblings

6T16, 10T32, 12T183 x 2, 15T28 x 2, 20T145, 20T149 x 2, 30T164 x 2, 30T166 x 2, 30T176 x 2, 36T1252, 40T589, 40T592 x 2, 45T96

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 $2^{3}$ $15$ $2$ $3$ $(1,3)(2,4)(5,6)$
2B $2,1^{4}$ $15$ $2$ $1$ $(3,5)$
2C $2^{2},1^{2}$ $45$ $2$ $2$ $(2,4)(3,6)$
3A $3^{2}$ $40$ $3$ $4$ $(1,2,6)(3,4,5)$
3B $3,1^{3}$ $40$ $3$ $2$ $(1,6,4)$
4A $4,2$ $90$ $4$ $4$ $(1,5)(2,3,4,6)$
4B $4,1^{2}$ $90$ $4$ $3$ $(2,5,6,3)$
5A $5,1$ $144$ $5$ $4$ $(1,6,4,3,2)$
6A $6$ $120$ $6$ $5$ $(1,5,2,3,6,4)$
6B $3,2,1$ $120$ $6$ $3$ $(1,4,6)(3,5)$

Malle's constant $a(G)$:     $1$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 3B 4A 4B 5A 6A 6B
Size 1 15 15 45 40 40 90 90 144 120 120
2 P 1A 1A 1A 1A 3A 3B 2C 2C 5A 3A 3B
3 P 1A 2A 2B 2C 1A 1A 4A 4B 5A 2A 2B
5 P 1A 2A 2B 2C 3A 3B 4A 4B 1A 6A 6B
Type
720.763.1a R 1 1 1 1 1 1 1 1 1 1 1
720.763.1b R 1 −1 −1 1 1 1 1 −1 1 −1 −1
720.763.5a R 5 −1 3 1 −1 2 −1 1 0 −1 0
720.763.5b R 5 3 −1 1 2 −1 −1 1 0 0 −1
720.763.5c R 5 −3 1 1 2 −1 −1 −1 0 0 1
720.763.5d R 5 1 −3 1 −1 2 −1 −1 0 1 0
720.763.9a R 9 3 3 1 0 0 1 −1 −1 0 0
720.763.9b R 9 −3 −3 1 0 0 1 1 −1 0 0
720.763.10a R 10 −2 2 −2 1 1 0 0 0 1 −1
720.763.10b R 10 2 −2 −2 1 1 0 0 0 −1 1
720.763.16a R 16 0 0 0 −2 −2 0 0 1 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{6}+s x^{4}+t x^{2}+u x+u$ Copy content Toggle raw display
The polynomial $f_{1}$ is generic for the base field $\Q$