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Properties

Label 21T40
Degree $21$
Order $6174$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^3:(C_3\times S_3)$

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Show commands: Magma

magma: G := TransitiveGroup(21, 40);
 

Group invariants

Abstract group:  $C_7^3:(C_3\times S_3)$
magma: IdentifyGroup(G);
 
Order:  $6174=2 \cdot 3^{2} \cdot 7^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $40$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,6,2)(4,5,7)(8,21,11,20,9,16)(10,18,12,15,13,17)(14,19)$, $(1,19,8,2,15,11,3,18,14,4,21,10,5,17,13,6,20,9,7,16,12)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$
$21$:  $C_7:C_3$
$42$:  $(C_7:C_3) \times C_2$
$126$:  21T11
$882$:  14T26

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: None

Low degree siblings

21T40 x 5, 42T464 x 6, 42T473 x 3, 42T474 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Conjugacy classes not computed

magma: ConjugacyClasses(G);
 

Character table

60 x 60 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed