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Properties

Label 21T13
21T13 1 4 1->4 11 1->11 2 5 2->5 8 2->8 3 6 3->6 12 3->12 7 4->7 9 4->9 5->1 13 5->13 6->2 10 6->10 7->3 14 7->14 8->9 21 8->21 9->10 17 9->17 10->11 20 10->20 11->12 16 11->16 12->13 19 12->19 13->14 15 13->15 14->8 18 14->18 15->5 15->19 16->1 16->20 17->4 17->21 18->7 18->15 19->3 19->16 20->6 20->17 21->2 21->18
Degree $21$
Order $147$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7:C_{21}$

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Copy content magma:G := TransitiveGroup(21, 13);
 

Group invariants

Abstract group:  $C_7:C_{21}$
Copy content magma:IdentifyGroup(G);
 
Order:  $147=3 \cdot 7^{2}$
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 $t$:  $13$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $7$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,4,7,3,6,2,5)(8,9,10,11,12,13,14)(15,19,16,20,17,21,18)$, $(1,11,16)(2,8,21)(3,12,19)(4,9,17)(5,13,15)(6,10,20)(7,14,18)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$
$7$:  $C_7$
$21$:  $C_7:C_3$, $C_{21}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T13

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{21}$ $1$ $1$ $0$ $()$
3A1 $3^{7}$ $7$ $3$ $14$ $( 1,16,11)( 2,21, 8)( 3,19,12)( 4,17, 9)( 5,15,13)( 6,20,10)( 7,18,14)$
3A-1 $3^{7}$ $7$ $3$ $14$ $( 1,11,16)( 2, 8,21)( 3,12,19)( 4, 9,17)( 5,13,15)( 6,10,20)( 7,14,18)$
7A1 $7^{3}$ $1$ $7$ $18$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$
7A-1 $7^{3}$ $1$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$
7A2 $7^{3}$ $1$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8, 9,10,11,12,13,14)(15,18,21,17,20,16,19)$
7A-2 $7^{3}$ $1$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$
7A3 $7^{3}$ $1$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$
7A-3 $7^{3}$ $1$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$
7B1 $7^{3}$ $3$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$
7B-1 $7^{3}$ $3$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,19,16,20,17,21,18)$
7C1 $7^{3}$ $3$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,19,16,20,17,21,18)$
7C-1 $7^{3}$ $3$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$
7C2 $7^{3}$ $3$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$
7C-2 $7^{3}$ $3$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)(15,16,17,18,19,20,21)$
7C3 $7^{3}$ $3$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8,11,14,10,13, 9,12)(15,20,18,16,21,19,17)$
7C-3 $7^{3}$ $3$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,14,13,12,11,10, 9)(15,20,18,16,21,19,17)$
7D1 $7^{2},1^{7}$ $3$ $7$ $12$ $( 8,10,12,14, 9,11,13)(15,17,19,21,16,18,20)$
7D-1 $7^{2},1^{7}$ $3$ $7$ $12$ $( 1, 2, 3, 4, 5, 6, 7)(15,16,17,18,19,20,21)$
7D2 $7^{2},1^{7}$ $3$ $7$ $12$ $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)$
7D-2 $7^{2},1^{7}$ $3$ $7$ $12$ $( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$
7D3 $7^{2},1^{7}$ $3$ $7$ $12$ $( 1, 6, 4, 2, 7, 5, 3)( 8,10,12,14, 9,11,13)$
7D-3 $7^{2},1^{7}$ $3$ $7$ $12$ $( 1, 4, 7, 3, 6, 2, 5)(15,18,21,17,20,16,19)$
21A1 $21$ $7$ $21$ $20$ $( 1,20, 9, 2,18,13, 3,16,10, 4,21,14, 5,19,11, 6,17, 8, 7,15,12)$
21A-1 $21$ $7$ $21$ $20$ $( 1,12,15, 7, 8,17, 6,11,19, 5,14,21, 4,10,16, 3,13,18, 2, 9,20)$
21A2 $21$ $7$ $21$ $20$ $( 1, 9,18, 3,10,21, 5,11,17, 7,12,20, 2,13,16, 4,14,19, 6, 8,15)$
21A-2 $21$ $7$ $21$ $20$ $( 1,15, 8, 6,19,14, 4,16,13, 2,20,12, 7,17,11, 5,21,10, 3,18, 9)$
21A4 $21$ $7$ $21$ $20$ $( 1,18,10, 5,17,12, 2,16,14, 6,15, 9, 3,21,11, 7,20,13, 4,19, 8)$
21A-4 $21$ $7$ $21$ $20$ $( 1, 8,19, 4,13,20, 7,11,21, 3, 9,15, 6,14,16, 2,12,17, 5,10,18)$
21A5 $21$ $7$ $21$ $20$ $( 1,13,21, 6,12,18, 4,11,15, 2,10,19, 7, 9,16, 5, 8,20, 3,14,17)$
21A-5 $21$ $7$ $21$ $20$ $( 1,17,14, 3,20, 8, 5,16, 9, 7,19,10, 2,15,11, 4,18,12, 6,21,13)$
21A8 $21$ $7$ $21$ $20$ $( 1,10,17, 2,14,15, 3,11,20, 4, 8,18, 5,12,16, 6, 9,21, 7,13,19)$
21A-8 $21$ $7$ $21$ $20$ $( 1,19,13, 7,21, 9, 6,16,12, 5,18, 8, 4,20,11, 3,15,14, 2,17,10)$
21A10 $21$ $7$ $21$ $20$ $( 1,21,12, 4,15,10, 7,16, 8, 3,17,13, 6,18,11, 2,19, 9, 5,20,14)$
21A-10 $21$ $7$ $21$ $20$ $( 1,14,20, 5, 9,19, 2,11,18, 6,13,17, 3, 8,16, 7,10,15, 4,12,21)$

Malle's constant $a(G)$:     $1/12$

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Character table

35 x 35 character table

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Regular extensions

Data not computed