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Properties

Label 42T33
42T33 1 25 1->25 39 1->39 2 26 2->26 40 2->40 3 28 3->28 37 3->37 4 27 4->27 38 4->38 5 30 5->30 42 5->42 6 29 6->29 41 6->41 7 31 7->31 7->31 8 32 8->32 8->32 9 34 9->34 35 9->35 10 33 10->33 36 10->36 11 11->34 11->36 12 12->33 12->35 13 13->30 13->41 14 14->29 14->42 15 15->28 15->38 16 16->27 16->37 17 17->26 17->40 18 18->25 18->39 19 19->2 20 20->1 21 21->4 24 21->24 22 22->3 23 22->23 23->6 23->21 24->5 24->22 25->10 25->17 26->9 26->18 27->11 27->15 28->12 28->16 29->7 30->8 31->8 31->14 32->7 32->13 33->11 33->15 34->12 34->16 35->17 36->18 37->24 38->23 39->2 39->20 40->1 40->19 41->5 41->22 42->6 42->21
Degree $42$
Order $168$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7:S_4$

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Copy content magma:G := TransitiveGroup(42, 33);
 

Group invariants

Abstract group:  $C_7:S_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $168=2^{3} \cdot 3 \cdot 7$
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 $t$:  $33$
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:  $(1,25,10,33,15,38,23,6,29,7,31,14,42,21,4,27,11,36,18,39,20)(2,26,9,34,16,37,24,5,30,8,32,13,41,22,3,28,12,35,17,40,19)$, $(1,39,2,40)(3,37)(4,38)(5,42,6,41)(7,31,8,32)(9,35)(10,36)(11,34,12,33)(13,30)(14,29)(15,28,16,27)(17,26,18,25)(21,24,22,23)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$14$:  $D_{7}$
$24$:  $S_4$
$42$:  $D_{21}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4$

Degree 7: $D_{7}$

Degree 14: None

Degree 21: $D_{21}$

Low degree siblings

28T30, 42T32

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

Malle's constant $a(G)$:     $1/14$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 3A 4A 7A1 7A2 7A3 14A1 14A3 14A5 21A1 21A2 21A4 21A5 21A8 21A10
Size 1 3 42 8 42 2 2 2 6 6 6 8 8 8 8 8 8
2 P 1A 1A 1A 3A 2A 7A2 7A3 7A1 7A1 7A3 7A2 21A2 21A4 21A8 21A10 21A5 21A1
3 P 1A 2A 2B 1A 4A 7A3 7A1 7A2 14A3 14A5 14A1 7A3 7A1 7A2 7A1 7A3 7A2
7 P 1A 2A 2B 3A 4A 1A 1A 1A 2A 2A 2A 3A 3A 3A 3A 3A 3A
Type
168.46.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
168.46.1b R 1 1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 1 1
168.46.2a R 2 2 0 −1 0 2 2 2 2 2 2 −1 −1 −1 −1 −1 −1
168.46.2b1 R 2 2 0 2 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73
168.46.2b2 R 2 2 0 2 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72
168.46.2b3 R 2 2 0 2 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7
168.46.2c1 R 2 2 0 −1 0 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−6+ζ216 ζ21−3+ζ213 ζ21−9+ζ219 ζ21−10+ζ2110 ζ21−4+ζ214 ζ21−8+ζ218 ζ21−2+ζ212 ζ21−1+ζ21 ζ21−5+ζ215
168.46.2c2 R 2 2 0 −1 0 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−6+ζ216 ζ21−3+ζ213 ζ21−9+ζ219 ζ21−4+ζ214 ζ21−10+ζ2110 ζ21−1+ζ21 ζ21−5+ζ215 ζ21−8+ζ218 ζ21−2+ζ212
168.46.2c3 R 2 2 0 −1 0 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−3+ζ213 ζ21−9+ζ219 ζ21−6+ζ216 ζ21−5+ζ215 ζ21−2+ζ212 ζ21−4+ζ214 ζ21−1+ζ21 ζ21−10+ζ2110 ζ21−8+ζ218
168.46.2c4 R 2 2 0 −1 0 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−3+ζ213 ζ21−9+ζ219 ζ21−6+ζ216 ζ21−2+ζ212 ζ21−5+ζ215 ζ21−10+ζ2110 ζ21−8+ζ218 ζ21−4+ζ214 ζ21−1+ζ21
168.46.2c5 R 2 2 0 −1 0 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−9+ζ219 ζ21−6+ζ216 ζ21−3+ζ213 ζ21−8+ζ218 ζ21−1+ζ21 ζ21−2+ζ212 ζ21−10+ζ2110 ζ21−5+ζ215 ζ21−4+ζ214
168.46.2c6 R 2 2 0 −1 0 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−9+ζ219 ζ21−6+ζ216 ζ21−3+ζ213 ζ21−1+ζ21 ζ21−8+ζ218 ζ21−5+ζ215 ζ21−4+ζ214 ζ21−2+ζ212 ζ21−10+ζ2110
168.46.3a R 3 −1 1 0 −1 3 3 3 −1 −1 −1 0 0 0 0 0 0
168.46.3b R 3 −1 −1 0 1 3 3 3 −1 −1 −1 0 0 0 0 0 0
168.46.6a1 R 6 −2 0 0 0 3ζ7−3+3ζ73 3ζ7−1+3ζ7 3ζ7−2+3ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 0 0 0 0 0 0
168.46.6a2 R 6 −2 0 0 0 3ζ7−2+3ζ72 3ζ7−3+3ζ73 3ζ7−1+3ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 0 0 0 0 0 0
168.46.6a3 R 6 −2 0 0 0 3ζ7−1+3ζ7 3ζ7−2+3ζ72 3ζ7−3+3ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed