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Properties

Label 42T5
42T5 1 9 1->9 17 1->17 2 10 2->10 18 2->18 3 12 3->12 16 3->16 4 11 4->11 15 4->15 5 8 5->8 13 5->13 6 7 6->7 14 6->14 7->8 7->13 8->14 9->12 9->16 10->11 10->15 11->18 12->17 21 13->21 22 14->22 23 15->23 24 16->24 19 17->19 20 18->20 25 19->25 40 19->40 26 20->26 39 20->39 27 21->27 38 21->38 28 22->28 37 22->37 30 23->30 41 23->41 29 24->29 42 24->42 33 25->33 35 25->35 34 26->34 36 26->36 27->33 27->35 28->34 28->36 31 29->31 32 29->32 30->31 30->32 31->41 32->42 33->38 34->37 35->40 36->39 37->6 38->5 39->2 40->1 41->4 42->3
Degree $42$
Order $42$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{21}$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(42, 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 $n$:  $42$
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);
 
$\card{\Aut(F/K)}$:  $42$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,17)(2,18)(3,16)(4,15)(5,13)(6,14)(7,8)(9,12)(10,11)(19,40)(20,39)(21,38)(22,37)(23,41)(24,42)(25,35)(26,36)(27,33)(28,34)(29,32)(30,31)$, $(1,9,16,24,29,31,41,4,11,18,20,26,34,37,6,7,13,21,27,35,40)(2,10,15,23,30,32,42,3,12,17,19,25,33,38,5,8,14,22,28,36,39)$
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 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 7: $D_{7}$

Degree 14: $D_{7}$

Degree 21: $D_{21}$

Low degree siblings

21T5

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{42}$ $1$ $1$ $0$ $()$
2A $2^{21}$ $21$ $2$ $21$ $( 1,17)( 2,18)( 3,16)( 4,15)( 5,13)( 6,14)( 7, 8)( 9,12)(10,11)(19,40)(20,39)(21,38)(22,37)(23,41)(24,42)(25,35)(26,36)(27,33)(28,34)(29,32)(30,31)$
3A $3^{14}$ $2$ $3$ $28$ $( 1, 4, 6)( 2, 3, 5)( 7, 9,11)( 8,10,12)(13,16,18)(14,15,17)(19,22,23)(20,21,24)(25,28,30)(26,27,29)(31,34,35)(32,33,36)(37,40,41)(38,39,42)$
7A1 $7^{6}$ $2$ $7$ $36$ $( 1,34,24, 7,41,27,18)( 2,33,23, 8,42,28,17)( 3,36,19,10,38,30,14)( 4,35,20, 9,37,29,13)( 5,32,22,12,39,25,15)( 6,31,21,11,40,26,16)$
7A2 $7^{6}$ $2$ $7$ $36$ $( 1,24,41,18,34, 7,27)( 2,23,42,17,33, 8,28)( 3,19,38,14,36,10,30)( 4,20,37,13,35, 9,29)( 5,22,39,15,32,12,25)( 6,21,40,16,31,11,26)$
7A3 $7^{6}$ $2$ $7$ $36$ $( 1, 7,18,24,27,34,41)( 2, 8,17,23,28,33,42)( 3,10,14,19,30,36,38)( 4, 9,13,20,29,35,37)( 5,12,15,22,25,32,39)( 6,11,16,21,26,31,40)$
21A1 $21^{2}$ $2$ $21$ $40$ $( 1,29,11,34,13,40,24, 4,26, 7,35,16,41,20, 6,27, 9,31,18,37,21)( 2,30,12,33,14,39,23, 3,25, 8,36,15,42,19, 5,28,10,32,17,38,22)$
21A2 $21^{2}$ $2$ $21$ $40$ $( 1,11,13,24,26,35,41, 6, 9,18,21,29,34,40, 4, 7,16,20,27,31,37)( 2,12,14,23,25,36,42, 5,10,17,22,30,33,39, 3, 8,15,19,28,32,38)$
21A4 $21^{2}$ $2$ $21$ $40$ $( 1,13,26,41, 9,21,34, 4,16,27,37,11,24,35, 6,18,29,40, 7,20,31)( 2,14,25,42,10,22,33, 3,15,28,38,12,23,36, 5,17,30,39, 8,19,32)$
21A5 $21^{2}$ $2$ $21$ $40$ $( 1,40,35,27,21,13, 7, 6,37,34,26,20,18,11, 4,41,31,29,24,16, 9)( 2,39,36,28,22,14, 8, 5,38,33,25,19,17,12, 3,42,32,30,23,15,10)$
21A8 $21^{2}$ $2$ $21$ $40$ $( 1,26, 9,34,16,37,24, 6,29, 7,31,13,41,21, 4,27,11,35,18,40,20)( 2,25,10,33,15,38,23, 5,30, 8,32,14,42,22, 3,28,12,36,17,39,19)$
21A10 $21^{2}$ $2$ $21$ $40$ $( 1,35,21, 7,37,26,18, 4,31,24, 9,40,27,13, 6,34,20,11,41,29,16)( 2,36,22, 8,38,25,17, 3,32,23,10,39,28,14, 5,33,19,12,42,30,15)$

Malle's constant $a(G)$:     $1/21$

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