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Properties

Label 42T13
42T13 1 27 1->27 2 28 2->28 3 5 3->5 26 3->26 4 6 4->6 25 4->25 30 5->30 29 6->29 7 34 7->34 37 7->37 8 33 8->33 38 8->38 9 32 9->32 42 9->42 10 31 10->31 41 10->41 11 36 11->36 40 11->40 12 35 12->35 39 12->39 13 13->31 13->39 14 14->32 14->40 15 15->35 15->38 16 16->36 16->37 17 17->33 17->41 18 18->34 18->42 19 19->4 19->25 20 20->3 20->26 21 21->1 21->30 22 22->2 22->29 23 23->6 23->28 24 24->5 24->27 25->10 26->9 27->7 28->8 29->12 30->11 31->16 32->15 33->14 34->13 35->18 36->17 37->21 38->22 39->19 40->20 41->23 42->24
Degree $42$
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(42, 13);
 

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

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 7: $D_{7}$

Degree 14: $D_{14}$

Degree 21: 21T8

Low degree siblings

21T8, 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 $1^{42}$ $1$ $1$ $0$ $()$
2A $2^{21}$ $3$ $2$ $21$ $( 1, 4)( 2, 3)( 5, 6)( 7,10)( 8, 9)(11,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,24)(25,27)(26,28)(29,30)(31,34)(32,33)(35,36)(37,39)(38,40)(41,42)$
2B $2^{21}$ $7$ $2$ $21$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,18)( 6,17)( 7, 8)( 9,10)(11,12)(19,38)(20,37)(21,40)(22,39)(23,41)(24,42)(25,32)(26,31)(27,33)(28,34)(29,36)(30,35)$
2C $2^{20},1^{2}$ $21$ $2$ $20$ $( 1,35)( 2,36)( 3,33)( 4,34)( 5,31)( 6,32)( 7,29)( 8,30)( 9,28)(10,27)(11,25)(12,26)(13,23)(14,24)(15,22)(16,21)(17,19)(18,20)(37,42)(38,41)$
3A $3^{14}$ $2$ $3$ $28$ $( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12)(13,15,17)(14,16,18)(19,22,23)(20,21,24)(25,28,29)(26,27,30)(31,33,35)(32,34,36)(37,40,42)(38,39,41)$
6A $6^{7}$ $14$ $6$ $35$ $( 1,16, 5,14, 3,18)( 2,15, 6,13, 4,17)( 7,10,11, 8, 9,12)(19,39,23,38,22,41)(20,40,24,37,21,42)(25,34,29,32,28,36)(26,33,30,31,27,35)$
7A1 $7^{6}$ $2$ $7$ $36$ $( 1, 7,13,19,25,31,37)( 2, 8,14,20,26,32,38)( 3, 9,15,22,28,33,40)( 4,10,16,21,27,34,39)( 5,11,17,23,29,35,42)( 6,12,18,24,30,36,41)$
7A2 $7^{6}$ $2$ $7$ $36$ $( 1,13,25,37, 7,19,31)( 2,14,26,38, 8,20,32)( 3,15,28,40, 9,22,33)( 4,16,27,39,10,21,34)( 5,17,29,42,11,23,35)( 6,18,30,41,12,24,36)$
7A3 $7^{6}$ $2$ $7$ $36$ $( 1,19,37,13,31, 7,25)( 2,20,38,14,32, 8,26)( 3,22,40,15,33, 9,28)( 4,21,39,16,34,10,27)( 5,23,42,17,35,11,29)( 6,24,41,18,36,12,30)$
14A1 $14^{3}$ $6$ $14$ $39$ $( 1,27, 7,34,13,39,19, 4,25,10,31,16,37,21)( 2,28, 8,33,14,40,20, 3,26, 9,32,15,38,22)( 5,30,11,36,17,41,23, 6,29,12,35,18,42,24)$
14A3 $14^{3}$ $6$ $14$ $39$ $( 1,32,19, 8,37,26,13, 2,31,20, 7,38,25,14)( 3,36,22,12,40,30,15, 6,33,24, 9,41,28,18)( 4,35,21,11,39,29,16, 5,34,23,10,42,27,17)$
14A5 $14^{3}$ $6$ $14$ $39$ $( 1,41,31,30,19,18, 7, 6,37,36,25,24,13,12)( 2,42,32,29,20,17, 8, 5,38,35,26,23,14,11)( 3,39,33,27,22,16, 9, 4,40,34,28,21,15,10)$
21A1 $21^{2}$ $4$ $21$ $40$ $( 1,40,35,25,22,17, 7, 3,42,31,28,23,13, 9, 5,37,33,29,19,15,11)( 2,39,36,26,21,18, 8, 4,41,32,27,24,14,10, 6,38,34,30,20,16,12)$
21A2 $21^{2}$ $4$ $21$ $40$ $( 1,35,22, 7,42,28,13, 5,33,19,11,40,25,17, 3,31,23, 9,37,29,15)( 2,36,21, 8,41,27,14, 6,34,20,12,39,26,18, 4,32,24,10,38,30,16)$
21A4 $21^{2}$ $4$ $21$ $40$ $( 1,22,42,13,33,11,25, 3,23,37,15,35, 7,28, 5,19,40,17,31, 9,29)( 2,21,41,14,34,12,26, 4,24,38,16,36, 8,27, 6,20,39,18,32,10,30)$

Malle's constant $a(G)$:     $1/20$

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