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Properties

Label 21T37
21T37 1 3 1->3 10 1->10 2 4 2->4 11 2->11 5 3->5 12 3->12 6 4->6 13 4->13 7 5->7 14 5->14 6->1 8 6->8 7->2 9 7->9 16 8->16 21 8->21 18 9->18 19 9->19 15 10->15 10->15 11->18 11->19 12->16 12->21 17 13->17 20 13->20 14->17 14->20 15->6 15->11 16->1 16->13 17->3 17->8 18->5 18->10 19->7 19->12 20->2 20->14 21->4 21->9
Degree $21$
Order $4116$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^3:D_6$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(21, 37);
 

Group invariants

Abstract group:  $C_7^3:D_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $4116=2^{2} \cdot 3 \cdot 7^{3}$
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 $t$:  $37$
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:  $(1,3,5,7,2,4,6)(8,21,9,18,10,15,11,19,12,16,13,20,14,17)$, $(1,10,15,6,8,16)(2,11,18,5,14,20)(3,12,21,4,13,17)(7,9,19)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$14$:  $D_{7}$
$28$:  $D_{14}$
$84$:  21T8
$588$:  14T25

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: None

Low degree siblings

21T37 x 5, 42T392 x 6, 42T393 x 6, 42T394 x 6, 42T400 x 3, 42T401 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

Copy content magma:ConjugacyClasses(G);
 

Character table

79 x 79 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed