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Properties

Label 21T8
21T8 1 2 1->2 17 1->17 3 2->3 16 2->16 3->1 18 3->18 4 20 4->20 4->20 5 19 5->19 21 5->21 6 6->19 6->21 7 7->2 7->17 8 8->1 8->18 9 9->3 9->16 10 10->5 14 10->14 11 11->4 15 11->15 12 12->6 13 12->13 13->8 13->11 14->7 14->12 15->9 15->10 16->8 16->11 17->9 17->10 18->7 18->12 19->5 19->14 20->6 20->13 21->4 21->15
Degree $21$
Order $84$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times D_7$

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Copy content magma:G := TransitiveGroup(21, 8);
 

Group invariants

Abstract group:  $S_3\times D_7$
Copy content magma:IdentifyGroup(G);
 
Order:  $84=2^{2} \cdot 3 \cdot 7$
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 $t$:  $8$
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:  $(1,2,3)(4,20,6,19,5,21)(7,17,9,16,8,18)(10,14,12,13,11,15)$, $(1,17,10,5,19,14,7,2,16,11,4,20,13,8)(3,18,12,6,21,15,9)$
Copy content magma:Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: $D_{7}$

Low degree siblings

42T13, 42T14, 42T15

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{21}$ $1$ $1$ $0$ $()$
2A $2^{7},1^{7}$ $3$ $2$ $7$ $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)$
2B $2^{9},1^{3}$ $7$ $2$ $9$ $( 1,16)( 2,17)( 3,18)( 4,13)( 5,14)( 6,15)( 7,10)( 8,11)( 9,12)$
2C $2^{10},1$ $21$ $2$ $10$ $( 1,11)( 2,10)( 3,12)( 4, 8)( 5, 7)( 6, 9)(13,20)(14,19)(15,21)(16,17)$
3A $3^{7}$ $2$ $3$ $14$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$
6A $6^{3},3$ $14$ $6$ $17$ $( 1,17, 3,16, 2,18)( 4,14, 6,13, 5,15)( 7,11, 9,10, 8,12)(19,20,21)$
7A1 $7^{3}$ $2$ $7$ $18$ $( 1,13, 4,16, 7,19,10)( 2,14, 5,17, 8,20,11)( 3,15, 6,18, 9,21,12)$
7A2 $7^{3}$ $2$ $7$ $18$ $( 1, 4, 7,10,13,16,19)( 2, 5, 8,11,14,17,20)( 3, 6, 9,12,15,18,21)$
7A3 $7^{3}$ $2$ $7$ $18$ $( 1,16,10, 4,19,13, 7)( 2,17,11, 5,20,14, 8)( 3,18,12, 6,21,15, 9)$
14A1 $14,7$ $6$ $14$ $19$ $( 1, 7,13,19, 4,10,16)( 2, 9,14,21, 5,12,17, 3, 8,15,20, 6,11,18)$
14A3 $14,7$ $6$ $14$ $19$ $( 1,21,16,15,10, 9, 4, 3,19,18,13,12, 7, 6)( 2,20,17,14,11, 8, 5)$
14A5 $14,7$ $6$ $14$ $19$ $( 1,11,19, 8,16, 5,13, 2,10,20, 7,17, 4,14)( 3,12,21, 9,18, 6,15)$
21A1 $21$ $4$ $21$ $20$ $( 1,11,21, 7,17, 6,13, 2,12,19, 8,18, 4,14, 3,10,20, 9,16, 5,15)$
21A2 $21$ $4$ $21$ $20$ $( 1,21,17,13,12, 8, 4, 3,20,16,15,11, 7, 6, 2,19,18,14,10, 9, 5)$
21A4 $21$ $4$ $21$ $20$ $( 1,17,12, 4,20,15, 7, 2,18,10, 5,21,13, 8, 3,16,11, 6,19,14, 9)$

Malle's constant $a(G)$:     $1/7$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 6A 7A1 7A2 7A3 14A1 14A3 14A5 21A1 21A2 21A4
Size 1 3 7 21 2 14 2 2 2 6 6 6 4 4 4
2 P 1A 1A 1A 1A 3A 3A 7A2 7A3 7A1 7A1 7A3 7A2 21A2 21A4 21A1
3 P 1A 2A 2B 2C 1A 2B 7A3 7A1 7A2 14A3 14A5 14A1 7A3 7A1 7A2
7 P 1A 2A 2B 2C 3A 6A 1A 1A 1A 2A 2A 2A 3A 3A 3A
Type
84.8.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
84.8.1b R 1 −1 −1 1 1 −1 1 1 1 −1 −1 −1 1 1 1
84.8.1c R 1 −1 1 −1 1 1 1 1 1 −1 −1 −1 1 1 1
84.8.1d R 1 1 −1 −1 1 −1 1 1 1 1 1 1 1 1 1
84.8.2a R 2 0 2 0 −1 −1 2 2 2 0 0 0 −1 −1 −1
84.8.2b R 2 0 −2 0 −1 1 2 2 2 0 0 0 −1 −1 −1
84.8.2c1 R 2 2 0 0 2 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7
84.8.2c2 R 2 2 0 0 2 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73
84.8.2c3 R 2 2 0 0 2 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72
84.8.2d1 R 2 −2 0 0 2 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7
84.8.2d2 R 2 −2 0 0 2 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73
84.8.2d3 R 2 −2 0 0 2 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72
84.8.4a1 R 4 0 0 0 −2 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 0 0 0 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7
84.8.4a2 R 4 0 0 0 −2 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 0 0 0 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73
84.8.4a3 R 4 0 0 0 −2 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 0 0 0 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed