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Properties

Label 21T5
21T5 1 8 1->8 17 1->17 2 7 2->7 16 2->16 3 9 3->9 18 3->18 4 6 4->6 14 4->14 5 13 5->13 15 6->15 11 7->11 10 8->10 12 9->12 19 10->19 21 11->21 20 12->20 13->17 14->16 15->18 19->20
Degree $21$
Order $42$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{21}$

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Copy content magma:G := TransitiveGroup(21, 5);
 

Group invariants

Abstract group:  $D_{21}$
Copy content magma:IdentifyGroup(G);
 
Order:  $42=2 \cdot 3 \cdot 7$
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 $t$:  $5$
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:  $(1,8)(2,7)(3,9)(4,6)(10,19)(11,21)(12,20)(13,17)(14,16)(15,18)$, $(1,17)(2,16)(3,18)(4,14)(5,13)(6,15)(7,11)(8,10)(9,12)(19,20)$
Copy content magma:Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: $D_{7}$

Low degree siblings

42T5

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 $2^{10},1$ $21$ $2$ $10$ $( 2, 3)( 4,21)( 5,20)( 6,19)( 7,16)( 8,18)( 9,17)(10,13)(11,15)(12,14)$
3A $3^{7}$ $2$ $3$ $14$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$
7A1 $7^{3}$ $2$ $7$ $18$ $( 1, 4, 9,12,14,17,21)( 2, 5, 7,10,15,18,19)( 3, 6, 8,11,13,16,20)$
7A2 $7^{3}$ $2$ $7$ $18$ $( 1, 9,14,21, 4,12,17)( 2, 7,15,19, 5,10,18)( 3, 8,13,20, 6,11,16)$
7A3 $7^{3}$ $2$ $7$ $18$ $( 1,12,21, 9,17, 4,14)( 2,10,19, 7,18, 5,15)( 3,11,20, 8,16, 6,13)$
21A1 $21$ $2$ $21$ $20$ $( 1,18,11, 4,19,13, 9, 2,16,12, 5,20,14, 7, 3,17,10, 6,21,15, 8)$
21A2 $21$ $2$ $21$ $20$ $( 1,11,19, 9,16, 5,14, 3,10,21, 8,18, 4,13, 2,12,20, 7,17, 6,15)$
21A4 $21$ $2$ $21$ $20$ $( 1,19,16,14,10, 8, 4, 2,20,17,15,11, 9, 5, 3,21,18,13,12, 7, 6)$
21A5 $21$ $2$ $21$ $20$ $( 1,13, 5,17, 8,19,12, 3,15, 4,16, 7,21,11, 2,14, 6,18, 9,20,10)$
21A8 $21$ $2$ $21$ $20$ $( 1,16,10, 4,20,15, 9, 3,18,12, 6,19,14, 8, 2,17,11, 5,21,13, 7)$
21A10 $21$ $2$ $21$ $20$ $( 1, 5, 8,12,15,16,21, 2, 6, 9,10,13,17,19, 3, 4, 7,11,14,18,20)$

Malle's constant $a(G)$:     $1/10$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A 7A1 7A2 7A3 21A1 21A2 21A4 21A5 21A8 21A10
Size 1 21 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 3A 7A2 7A3 7A1 21A2 21A4 21A8 21A10 21A5 21A1
3 P 1A 2A 1A 7A3 7A1 7A2 7A1 7A2 7A3 7A2 7A1 7A3
7 P 1A 2A 3A 1A 1A 1A 3A 3A 3A 3A 3A 3A
Type
42.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1
42.5.1b R 1 −1 1 1 1 1 1 1 1 1 1 1
42.5.2a R 2 0 −1 2 2 2 −1 −1 −1 −1 −1 −1
42.5.2b1 R 2 0 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72
42.5.2b2 R 2 0 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7
42.5.2b3 R 2 0 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73
42.5.2c1 R 2 0 −1 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−8+ζ218 ζ21−10+ζ2110 ζ21−5+ζ215 ζ21−2+ζ212 ζ21−4+ζ214 ζ21−1+ζ21
42.5.2c2 R 2 0 −1 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−1+ζ21 ζ21−4+ζ214 ζ21−2+ζ212 ζ21−5+ζ215 ζ21−10+ζ2110 ζ21−8+ζ218
42.5.2c3 R 2 0 −1 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−10+ζ2110 ζ21−2+ζ212 ζ21−1+ζ21 ζ21−8+ζ218 ζ21−5+ζ215 ζ21−4+ζ214
42.5.2c4 R 2 0 −1 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−4+ζ214 ζ21−5+ζ215 ζ21−8+ζ218 ζ21−1+ζ21 ζ21−2+ζ212 ζ21−10+ζ2110
42.5.2c5 R 2 0 −1 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−5+ζ215 ζ21−1+ζ21 ζ21−10+ζ2110 ζ21−4+ζ214 ζ21−8+ζ218 ζ21−2+ζ212
42.5.2c6 R 2 0 −1 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−2+ζ212 ζ21−8+ζ218 ζ21−4+ζ214 ζ21−10+ζ2110 ζ21−1+ζ21 ζ21−5+ζ215

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed