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Properties

Label 21T46
21T46 1 5 1->5 8 1->8 2 4 2->4 12 2->12 3 9 3->9 13 4->13 10 5->10 6 7 6->7 14 6->14 11 7->11 16 8->16 20 8->20 17 9->17 19 9->19 15 10->15 21 10->21 18 11->18 11->18 12->15 12->21 13->17 13->19 14->16 14->20 15->1 15->14 16->5 16->9 17->2 17->11 18->6 18->13 19->3 19->8 20->7 20->10 21->4
Degree $21$
Order $8232$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^3:S_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(21, 46);
 

Group invariants

Abstract group:  $C_7^3:S_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $8232=2^{3} \cdot 3 \cdot 7^{3}$
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 $n$:  $21$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $46$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,8,20,7,11,18,6,14,16,5,10,21,4,13,19,3,9,17,2,12,15)$, $(1,5)(2,4)(6,7)(8,16,9,19)(10,15,14,20)(11,18,13,17)(12,21)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$24$:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: None

Low degree siblings

28T347, 42T538, 42T539, 42T540, 42T548

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

Copy content magma:ConjugacyClasses(G);
 

Character table

70 x 70 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed