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Properties

Label 42T34
42T34 1 5 1->5 37 1->37 2 6 2->6 38 2->38 3 3->2 41 3->41 4 4->1 42 4->42 5->4 40 5->40 6->3 39 6->39 7 7->1 11 7->11 8 8->2 12 8->12 9 9->6 9->7 10 10->5 10->8 11->3 11->9 12->4 12->10 13 13->8 17 13->17 14 14->7 18 14->18 15 15->11 15->13 16 16->12 16->14 17->9 17->15 18->10 18->16 19 19->14 24 19->24 20 20->13 23 20->23 21 21->18 21->20 22 22->17 22->19 23->16 23->21 24->15 24->22 25 25->19 30 25->30 26 26->20 29 26->29 27 27->24 27->26 28 28->23 28->25 29->22 29->27 30->21 30->28 31 31->26 35 31->35 32 32->25 36 32->36 33 33->30 33->31 34 34->29 34->32 35->28 35->33 36->27 36->34 37->31 37->41 38->32 38->42 39->36 39->37 40->35 40->38 41->34 41->39 42->33 42->40
Degree $42$
Order $168$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_7\times S_4$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(42, 34);
 
Copy content sage:G = TransitiveGroup(42, 34)
 
Copy content oscar:G = transitive_group(42, 34)
 
Copy content gap:G := TransitiveGroup(42, 34);
 

Group invariants

Abstract group:  $C_7\times S_4$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Copy content oscar:small_group_identification(G)
 
Copy content gap:IdGroup(G);
 
Order:  $168=2^{3} \cdot 3 \cdot 7$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Copy content gap:Order(G);
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content gap:IsCyclic(G);
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Copy content gap:IsAbelian(G);
 
Solvable:  yes
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Copy content gap:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 

Group action invariants

Degree $n$:  $42$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Copy content gap:NrMovedPoints(G);
 
Transitive number $t$:  $34$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
Copy content gap:TransitiveIdentification(G);
 
Parity:  $-1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Copy content gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
 
Transitivity:  1
Primitive:  no
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
Copy content gap:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $14$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(42).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(42), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(42), G));
 
Generators:  $(1,5,4)(2,6,3)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,24,22)(20,23,21)(25,30,28)(26,29,27)(31,35,33)(32,36,34)(37,41,39)(38,42,40)$, $(1,37,31,26,20,13,8,2,38,32,25,19,14,7)(3,41,34,29,22,17,9,6,39,36,27,24,15,11)(4,42,33,30,21,18,10,5,40,35,28,23,16,12)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$7$:  $C_7$
$14$:  $C_{14}$
$24$:  $S_4$
$42$:  21T6

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4$

Degree 7: $C_7$

Degree 14: None

Degree 21: 21T6

Low degree siblings

28T31, 42T35

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 $2^{14},1^{14}$ $3$ $2$ $14$ $( 3, 4)( 5, 6)( 9,10)(11,12)(15,16)(17,18)(21,22)(23,24)(27,28)(29,30)(33,34)(35,36)(39,40)(41,42)$
2B $2^{21}$ $6$ $2$ $21$ $( 1, 2)( 3, 6)( 4, 5)( 7, 8)( 9,11)(10,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,24)(25,26)(27,29)(28,30)(31,32)(33,35)(34,36)(37,38)(39,41)(40,42)$
3A $3^{14}$ $8$ $3$ $28$ $( 1, 5, 3)( 2, 6, 4)( 7,11,10)( 8,12, 9)(13,17,16)(14,18,15)(19,24,21)(20,23,22)(25,30,27)(26,29,28)(31,35,34)(32,36,33)(37,41,40)(38,42,39)$
4A $4^{7},1^{14}$ $6$ $4$ $21$ $( 3, 6, 4, 5)( 9,11,10,12)(15,17,16,18)(21,23,22,24)(27,29,28,30)(33,35,34,36)(39,41,40,42)$
7A1 $7^{6}$ $1$ $7$ $36$ $( 1,20,38,14,31, 8,25)( 2,19,37,13,32, 7,26)( 3,22,39,15,34, 9,27)( 4,21,40,16,33,10,28)( 5,23,42,18,35,12,30)( 6,24,41,17,36,11,29)$
7A-1 $7^{6}$ $1$ $7$ $36$ $( 1,25, 8,31,14,38,20)( 2,26, 7,32,13,37,19)( 3,27, 9,34,15,39,22)( 4,28,10,33,16,40,21)( 5,30,12,35,18,42,23)( 6,29,11,36,17,41,24)$
7A2 $7^{6}$ $1$ $7$ $36$ $( 1,38,31,25,20,14, 8)( 2,37,32,26,19,13, 7)( 3,39,34,27,22,15, 9)( 4,40,33,28,21,16,10)( 5,42,35,30,23,18,12)( 6,41,36,29,24,17,11)$
7A-2 $7^{6}$ $1$ $7$ $36$ $( 1, 8,14,20,25,31,38)( 2, 7,13,19,26,32,37)( 3, 9,15,22,27,34,39)( 4,10,16,21,28,33,40)( 5,12,18,23,30,35,42)( 6,11,17,24,29,36,41)$
7A3 $7^{6}$ $1$ $7$ $36$ $( 1,14,25,38, 8,20,31)( 2,13,26,37, 7,19,32)( 3,15,27,39, 9,22,34)( 4,16,28,40,10,21,33)( 5,18,30,42,12,23,35)( 6,17,29,41,11,24,36)$
7A-3 $7^{6}$ $1$ $7$ $36$ $( 1,31,20, 8,38,25,14)( 2,32,19, 7,37,26,13)( 3,34,22, 9,39,27,15)( 4,33,21,10,40,28,16)( 5,35,23,12,42,30,18)( 6,36,24,11,41,29,17)$
14A1 $14^{2},7^{2}$ $3$ $14$ $38$ $( 1,20,38,14,31, 8,25)( 2,19,37,13,32, 7,26)( 3,21,39,16,34,10,27, 4,22,40,15,33, 9,28)( 5,24,42,17,35,11,30, 6,23,41,18,36,12,29)$
14A-1 $14^{2},7^{2}$ $3$ $14$ $38$ $( 1,25, 8,31,14,38,20)( 2,26, 7,32,13,37,19)( 3,28, 9,33,15,40,22, 4,27,10,34,16,39,21)( 5,29,12,36,18,41,23, 6,30,11,35,17,42,24)$
14A3 $14^{2},7^{2}$ $3$ $14$ $38$ $( 1,14,25,38, 8,20,31)( 2,13,26,37, 7,19,32)( 3,16,27,40, 9,21,34, 4,15,28,39,10,22,33)( 5,17,30,41,12,24,35, 6,18,29,42,11,23,36)$
14A-3 $14^{2},7^{2}$ $3$ $14$ $38$ $( 1,31,20, 8,38,25,14)( 2,32,19, 7,37,26,13)( 3,33,22,10,39,28,15, 4,34,21, 9,40,27,16)( 5,36,23,11,42,29,18, 6,35,24,12,41,30,17)$
14A5 $14^{2},7^{2}$ $3$ $14$ $38$ $( 1, 8,14,20,25,31,38)( 2, 7,13,19,26,32,37)( 3,10,15,21,27,33,39, 4, 9,16,22,28,34,40)( 5,11,18,24,30,36,42, 6,12,17,23,29,35,41)$
14A-5 $14^{2},7^{2}$ $3$ $14$ $38$ $( 1,38,31,25,20,14, 8)( 2,37,32,26,19,13, 7)( 3,40,34,28,22,16, 9, 4,39,33,27,21,15,10)( 5,41,35,29,23,17,12, 6,42,36,30,24,18,11)$
14B1 $14^{3}$ $6$ $14$ $39$ $( 1,32,20, 7,38,26,14, 2,31,19, 8,37,25,13)( 3,36,22,11,39,29,15, 6,34,24, 9,41,27,17)( 4,35,21,12,40,30,16, 5,33,23,10,42,28,18)$
14B-1 $14^{3}$ $6$ $14$ $39$ $( 1,13,25,37, 8,19,31, 2,14,26,38, 7,20,32)( 3,17,27,41, 9,24,34, 6,15,29,39,11,22,36)( 4,18,28,42,10,23,33, 5,16,30,40,12,21,35)$
14B3 $14^{3}$ $6$ $14$ $39$ $( 1, 7,14,19,25,32,38, 2, 8,13,20,26,31,37)( 3,11,15,24,27,36,39, 6, 9,17,22,29,34,41)( 4,12,16,23,28,35,40, 5,10,18,21,30,33,42)$
14B-3 $14^{3}$ $6$ $14$ $39$ $( 1,37,31,26,20,13, 8, 2,38,32,25,19,14, 7)( 3,42,34,30,22,18, 9, 5,39,35,27,23,15,12)( 4,41,33,29,21,17,10, 6,40,36,28,24,16,11)$
14B5 $14^{3}$ $6$ $14$ $39$ $( 1,26, 8,32,14,37,20, 2,25, 7,31,13,38,19)( 3,29, 9,36,15,41,22, 6,27,11,34,17,39,24)( 4,30,10,35,16,42,21, 5,28,12,33,18,40,23)$
14B-5 $14^{3}$ $6$ $14$ $39$ $( 1,19,38,13,31, 7,25, 2,20,37,14,32, 8,26)( 3,24,39,17,34,11,27, 6,22,41,15,36, 9,29)( 4,23,40,18,33,12,28, 5,21,42,16,35,10,30)$
21A1 $21^{2}$ $8$ $21$ $40$ $( 1,35,22, 8,42,27,14, 5,34,20,12,39,25,18, 3,31,23, 9,38,30,15)( 2,36,21, 7,41,28,13, 6,33,19,11,40,26,17, 4,32,24,10,37,29,16)$
21A-1 $21^{2}$ $8$ $21$ $40$ $( 1,18,27,38,12,22,31, 5,15,25,42, 9,20,35, 3,14,30,39, 8,23,34)( 2,17,28,37,11,21,32, 6,16,26,41,10,19,36, 4,13,29,40, 7,24,33)$
21A2 $21^{2}$ $8$ $21$ $40$ $( 1,23,39,14,35, 9,25, 5,22,38,18,34, 8,30, 3,20,42,15,31,12,27)( 2,24,40,13,36,10,26, 6,21,37,17,33, 7,29, 4,19,41,16,32,11,28)$
21A-2 $21^{2}$ $8$ $21$ $40$ $( 1,30, 9,31,18,39,20, 5,27, 8,35,15,38,23, 3,25,12,34,14,42,22)( 2,29,10,32,17,40,19, 6,28, 7,36,16,37,24, 4,26,11,33,13,41,21)$
21A4 $21^{2}$ $8$ $21$ $40$ $( 1,41,34,25,24,15, 8, 6,39,31,29,22,14,11, 3,38,36,27,20,17, 9)( 2,42,33,26,23,16, 7, 5,40,32,30,21,13,12, 4,37,35,28,19,18,10)$
21A-4 $21^{2}$ $8$ $21$ $40$ $( 1,12,15,20,30,34,38, 5, 9,14,23,27,31,42, 3, 8,18,22,25,35,39)( 2,11,16,19,29,33,37, 6,10,13,24,28,32,41, 4, 7,17,21,26,36,40)$
28A1 $28,7^{2}$ $6$ $28$ $39$ $( 1,31,20, 8,38,25,14)( 2,32,19, 7,37,26,13)( 3,35,21,11,39,30,16, 6,34,23,10,41,27,18, 4,36,22,12,40,29,15, 5,33,24, 9,42,28,17)$
28A-1 $28,7^{2}$ $6$ $28$ $39$ $( 1,14,25,38, 8,20,31)( 2,13,26,37, 7,19,32)( 3,18,28,41, 9,23,33, 6,15,30,40,11,22,35, 4,17,27,42,10,24,34, 5,16,29,39,12,21,36)$
28A3 $28,7^{2}$ $6$ $28$ $39$ $( 1, 8,14,20,25,31,38)( 2, 7,13,19,26,32,37)( 3,12,16,24,27,35,40, 6, 9,18,21,29,34,42, 4,11,15,23,28,36,39, 5,10,17,22,30,33,41)$
28A-3 $28,7^{2}$ $6$ $28$ $39$ $( 1,38,31,25,20,14, 8)( 2,37,32,26,19,13, 7)( 3,41,33,30,22,17,10, 5,39,36,28,23,15,11, 4,42,34,29,21,18, 9, 6,40,35,27,24,16,12)$
28A5 $28,7^{2}$ $6$ $28$ $39$ $( 1,25, 8,31,14,38,20)( 2,26, 7,32,13,37,19)( 3,30,10,36,15,42,21, 6,27,12,33,17,39,23, 4,29, 9,35,16,41,22, 5,28,11,34,18,40,24)$
28A-5 $28,7^{2}$ $6$ $28$ $39$ $( 1,20,38,14,31, 8,25)( 2,19,37,13,32, 7,26)( 3,23,40,17,34,12,28, 6,22,42,16,36, 9,30, 4,24,39,18,33,11,27, 5,21,41,15,35,10,29)$

Malle's constant $a(G)$:     $1/14$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

35 x 35 character table

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 
Copy content gap:CharacterTable(G);
 

Regular extensions

Data not computed