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Properties

Label 42T61
42T61 1 2 1->2 42 1->42 3 2->3 40 2->40 3->1 41 3->41 4 38 4->38 4->42 5 39 5->39 5->41 6 37 6->37 6->40 7 12 7->12 13 7->13 8 10 8->10 14 8->14 9 11 9->11 15 9->15 10->9 10->12 11->7 11->10 12->8 12->11 23 13->23 26 13->26 22 14->22 25 14->25 24 15->24 27 15->27 16 21 16->21 16->22 17 19 17->19 17->24 18 20 18->20 18->23 35 19->35 19->39 36 20->36 20->37 34 21->34 21->38 32 22->32 22->34 33 23->33 23->36 31 24->31 24->35 25->6 25->8 26->5 26->7 27->4 27->9 28 28->1 28->4 29 29->2 29->6 30 30->3 30->5 31->17 31->20 32->16 32->19 33->18 33->21 34->14 34->16 35->15 35->17 36->13 36->18 37->29 37->31 38->28 38->33 39->30 39->32 40->25 40->29 41->26 41->30 42->27 42->28
Degree $42$
Order $294$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7:F_7$

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Copy content magma:G := TransitiveGroup(42, 61);
 

Group invariants

Abstract group:  $C_7:F_7$
Copy content magma:IdentifyGroup(G);
 
Order:  $294=2 \cdot 3 \cdot 7^{2}$
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 $t$:  $61$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $21$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,42,27,4,38,28)(2,40,25,6,37,29)(3,41,26,5,39,30)(7,12,8,10,9,11)(13,23,33,18,20,36)(14,22,32,16,21,34)(15,24,31,17,19,35)$, $(1,2,3)(4,42,28)(5,41,30)(6,40,29)(7,13,26)(8,14,25)(9,15,27)(10,12,11)(16,22,34)(17,24,35)(18,23,36)(19,39,32)(20,37,31)(21,38,33)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$21$:  $C_7:C_3$
$42$:  $F_7$, $(C_7:C_3) \times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 7: None

Degree 14: 14T14

Degree 21: None

Low degree siblings

14T14 x 3, 42T61 x 2

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 $2^{21}$ $7$ $2$ $21$ $( 1,29)( 2,30)( 3,28)( 4,21)( 5,19)( 6,20)( 7,36)( 8,34)( 9,35)(10,27)(11,26)(12,25)(13,41)(14,40)(15,42)(16,33)(17,32)(18,31)(22,39)(23,38)(24,37)$
3A1 $3^{14}$ $49$ $3$ $28$ $( 1, 2, 3)( 4,23,16)( 5,22,17)( 6,24,18)( 7,13,26)( 8,14,25)( 9,15,27)(10,35,42)(11,36,41)(12,34,40)(19,39,32)(20,37,31)(21,38,33)(28,29,30)$
3A-1 $3^{14}$ $49$ $3$ $28$ $( 1, 3, 2)( 4,16,23)( 5,17,22)( 6,18,24)( 7,26,13)( 8,25,14)( 9,27,15)(10,42,35)(11,41,36)(12,40,34)(19,32,39)(20,31,37)(21,33,38)(28,30,29)$
6A1 $6^{7}$ $49$ $6$ $35$ $( 1,28, 2,29, 3,30)( 4,33,23,21,16,38)( 5,32,22,19,17,39)( 6,31,24,20,18,37)( 7,11,13,36,26,41)( 8,12,14,34,25,40)( 9,10,15,35,27,42)$
6A-1 $6^{7}$ $49$ $6$ $35$ $( 1,30, 3,29, 2,28)( 4,38,16,21,23,33)( 5,39,17,19,22,32)( 6,37,18,20,24,31)( 7,41,26,36,13,11)( 8,40,25,34,14,12)( 9,42,27,35,15,10)$
7A1 $7^{6}$ $3$ $7$ $36$ $( 1,21,39,13,31, 9,25)( 2,20,38,15,32, 8,26)( 3,19,37,14,33, 7,27)( 4,22,41,18,35,12,29)( 5,24,40,16,36,10,28)( 6,23,42,17,34,11,30)$
7A-1 $7^{6}$ $3$ $7$ $36$ $( 1,25, 9,31,13,39,21)( 2,26, 8,32,15,38,20)( 3,27, 7,33,14,37,19)( 4,29,12,35,18,41,22)( 5,28,10,36,16,40,24)( 6,30,11,34,17,42,23)$
7B $7^{6}$ $6$ $7$ $36$ $( 1,13,25,39, 9,21,31)( 2,15,26,38, 8,20,32)( 3,14,27,37, 7,19,33)( 4,35,22,12,41,29,18)( 5,36,24,10,40,28,16)( 6,34,23,11,42,30,17)$
7C1 $7^{3},1^{21}$ $6$ $7$ $18$ $( 1,25, 9,31,13,39,21)( 2,26, 8,32,15,38,20)( 3,27, 7,33,14,37,19)$
7C-1 $7^{3},1^{21}$ $6$ $7$ $18$ $( 1,31,21, 9,39,25,13)( 2,32,20, 8,38,26,15)( 3,33,19, 7,37,27,14)$
7D1 $7^{6}$ $6$ $7$ $36$ $( 1,39,31,25,21,13, 9)( 2,38,32,26,20,15, 8)( 3,37,33,27,19,14, 7)( 4,35,22,12,41,29,18)( 5,36,24,10,40,28,16)( 6,34,23,11,42,30,17)$
7D-1 $7^{6}$ $6$ $7$ $36$ $( 1, 9,13,21,25,31,39)( 2, 8,15,20,26,32,38)( 3, 7,14,19,27,33,37)( 4,29,12,35,18,41,22)( 5,28,10,36,16,40,24)( 6,30,11,34,17,42,23)$
7E1 $7^{6}$ $6$ $7$ $36$ $( 1,39,31,25,21,13, 9)( 2,38,32,26,20,15, 8)( 3,37,33,27,19,14, 7)( 4,18,29,41,12,22,35)( 5,16,28,40,10,24,36)( 6,17,30,42,11,23,34)$
7E-1 $7^{6}$ $6$ $7$ $36$ $( 1,21,39,13,31, 9,25)( 2,20,38,15,32, 8,26)( 3,19,37,14,33, 7,27)( 4,18,29,41,12,22,35)( 5,16,28,40,10,24,36)( 6,17,30,42,11,23,34)$
14A1 $14^{3}$ $21$ $14$ $39$ $( 1,18,21,35,39,12,13,29,31, 4, 9,22,25,41)( 2,17,20,34,38,11,15,30,32, 6, 8,23,26,42)( 3,16,19,36,37,10,14,28,33, 5, 7,24,27,40)$
14A-1 $14^{3}$ $21$ $14$ $39$ $( 1,41,25,22, 9, 4,31,29,13,12,39,35,21,18)( 2,42,26,23, 8, 6,32,30,15,11,38,34,20,17)( 3,40,27,24, 7, 5,33,28,14,10,37,36,19,16)$

Malle's constant $a(G)$:     $1/18$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 6A1 6A-1 7A1 7A-1 7B 7C1 7C-1 7D1 7D-1 7E1 7E-1 14A1 14A-1
Size 1 7 49 49 49 49 3 3 6 6 6 6 6 6 6 21 21
2 P 1A 1A 3A-1 3A1 3A1 3A-1 7A1 7A-1 7B 7C1 7C-1 7D1 7D-1 7E1 7E-1 7A1 7A-1
3 P 1A 2A 1A 1A 2A 2A 7A-1 7A1 7B 7C-1 7C1 7D-1 7D1 7E-1 7E1 14A-1 14A1
7 P 1A 2A 3A1 3A-1 6A1 6A-1 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A
Type
294.12.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
294.12.1b R 1 −1 1 1 −1 −1 1 1 1 1 1 1 1 1 1 −1 −1
294.12.1c1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 1 1 1 1 1 1
294.12.1c2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 1 1 1 1 1 1
294.12.1d1 C 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 1 1 1 1 1 1 −1 −1
294.12.1d2 C 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 1 1 1 1 1 1 −1 −1
294.12.3a1 C 3 3 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 3 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
294.12.3a2 C 3 3 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 3 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
294.12.3b1 C 3 −3 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 3 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72
294.12.3b2 C 3 −3 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 3 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72
294.12.6a R 6 0 0 0 0 0 6 6 −1 −1 −1 −1 −1 −1 −1 0 0
294.12.6b1 C 6 0 0 0 0 0 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 −1 2ζ7−3+2ζ7+2ζ72 −1 −1 −ζ7−3+2−ζ7−ζ72 ζ7−3+3+ζ7+ζ72 0 0
294.12.6b2 C 6 0 0 0 0 0 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 −1 −2ζ7−3−2−2ζ7−2ζ72 −1 −1 ζ7−3+3+ζ7+ζ72 −ζ7−3+2−ζ7−ζ72 0 0
294.12.6c1 C 6 0 0 0 0 0 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 −1 −1 −1 ζ7−3+3+ζ7+ζ72 −ζ7−3+2−ζ7−ζ72 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 0 0
294.12.6c2 C 6 0 0 0 0 0 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 −1 −1 −1 −ζ7−3+2−ζ7−ζ72 ζ7−3+3+ζ7+ζ72 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 0 0
294.12.6d1 C 6 0 0 0 0 0 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 ζ7−3+3+ζ7+ζ72 −1 −ζ7−3+2−ζ7−ζ72 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 −1 −1 0 0
294.12.6d2 C 6 0 0 0 0 0 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 −ζ7−3+2−ζ7−ζ72 −1 ζ7−3+3+ζ7+ζ72 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 −1 −1 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed