Properties

Label 42T263
42T263 1 12 1->12 15 1->15 2 11 2->11 16 2->16 3 10 3->10 14 3->14 4 9 4->9 13 4->13 5 7 5->7 17 5->17 6 8 6->8 18 6->18 7->4 36 7->36 8->3 35 8->35 9->2 33 9->33 10->1 34 10->34 11->6 32 11->32 12->5 31 12->31 23 13->23 27 13->27 24 14->24 28 14->28 21 15->21 26 15->26 22 16->22 25 16->25 19 17->19 30 17->30 20 18->20 29 18->29 40 19->40 41 19->41 39 20->39 42 20->42 38 21->38 21->40 37 22->37 22->39 23->37 23->41 24->38 24->42 25->18 25->33 26->17 26->34 27->16 27->32 28->15 28->31 29->14 29->36 30->13 30->35 31->21 31->29 32->22 32->30 33->20 33->27 34->19 34->28 35->23 35->26 36->24 36->25 37->5 37->9 38->6 38->10 39->4 39->7 40->3 40->8 41->1 41->11 42->2 42->12
Degree $42$
Order $2016$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $D_6\times \GL(3,2)$

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, 263);
 
Copy content sage:G = TransitiveGroup(42, 263)
 
Copy content oscar:G = transitive_group(42, 263)
 
Copy content gap:G := TransitiveGroup(42, 263);
 

Group invariants

Abstract group:  $D_6\times \GL(3,2)$
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:  $2016=2^{5} \cdot 3^{2} \cdot 7$
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:  no
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 $t$:  $263$
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);
 
$\card{\Aut(F/K)}$:  $2$
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:  $(1,12,31,29,14,24,38,6,8,35,26,17,19,41)(2,11,32,30,13,23,37,5,7,36,25,18,20,42)(3,10,34,28,15,21,40)(4,9,33,27,16,22,39)$, $(1,15,26,34,19,40,8,3,14,28,31,21,38,10)(2,16,25,33,20,39,7,4,13,27,32,22,37,9)(5,17,30,35,23,41,11,6,18,29,36,24,42,12)$
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)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$168$:  $\GL(3,2)$
$336$:  14T17 x 3
$672$:  28T84
$1008$:  21T27

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 7: $\GL(3,2)$

Degree 14: $\GL(3,2) \times C_2$

Degree 21: 21T27

Low degree siblings

42T263 x 3, 42T264 x 4

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{42}$ $1$ $1$ $0$ $()$
2A $2^{21}$ $1$ $2$ $21$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)$
2B $2^{14},1^{14}$ $3$ $2$ $14$ $( 1, 6)( 2, 5)( 7,11)( 8,12)(13,18)(14,17)(19,24)(20,23)(25,30)(26,29)(31,35)(32,36)(37,42)(38,41)$
2C $2^{21}$ $3$ $2$ $21$ $( 1, 5)( 2, 6)( 3, 4)( 7,12)( 8,11)( 9,10)(13,17)(14,18)(15,16)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,36)(32,35)(33,34)(37,41)(38,42)(39,40)$
2D $2^{21}$ $21$ $2$ $21$ $( 1, 2)( 3, 4)( 5, 6)( 7,26)( 8,25)( 9,28)(10,27)(11,29)(12,30)(13,14)(15,16)(17,18)(19,32)(20,31)(21,33)(22,34)(23,35)(24,36)(37,38)(39,40)(41,42)$
2E $2^{12},1^{18}$ $21$ $2$ $12$ $( 7,32)( 8,31)( 9,33)(10,34)(11,36)(12,35)(19,26)(20,25)(21,28)(22,27)(23,30)(24,29)$
2F $2^{21}$ $63$ $2$ $21$ $( 1,42)( 2,41)( 3,39)( 4,40)( 5,38)( 6,37)( 7,12)( 8,11)( 9,10)(13,17)(14,18)(15,16)(19,36)(20,35)(21,33)(22,34)(23,31)(24,32)(25,29)(26,30)(27,28)$
2G $2^{18},1^{6}$ $63$ $2$ $18$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 7)( 6, 8)(13,36)(14,35)(15,34)(16,33)(17,31)(18,32)(19,24)(20,23)(25,30)(26,29)(37,42)(38,41)$
3A $3^{14}$ $2$ $3$ $28$ $( 1, 4, 6)( 2, 3, 5)( 7,10,11)( 8, 9,12)(13,15,18)(14,16,17)(19,22,24)(20,21,23)(25,28,30)(26,27,29)(31,33,35)(32,34,36)(37,40,42)(38,39,41)$
3B $3^{12},1^{6}$ $56$ $3$ $24$ $( 1,19, 8)( 2,20, 7)( 3,21,10)( 4,22, 9)( 5,23,11)( 6,24,12)(13,32,37)(14,31,38)(15,34,40)(16,33,39)(17,35,41)(18,36,42)$
3C $3^{14}$ $112$ $3$ $28$ $( 1,22,12)( 2,21,11)( 3,23, 7)( 4,24, 8)( 5,20,10)( 6,19, 9)(13,40,30)(14,39,29)(15,42,25)(16,41,26)(17,38,27)(18,37,28)(31,33,35)(32,34,36)$
4A $4^{6},2^{6},1^{6}$ $42$ $4$ $24$ $( 1,14)( 2,13)( 3,15)( 4,16)( 5,18)( 6,17)( 7,25,32,20)( 8,26,31,19)( 9,27,33,22)(10,28,34,21)(11,30,36,23)(12,29,35,24)$
4B $4^{6},2^{9}$ $42$ $4$ $27$ $( 1,13, 8,32)( 2,14, 7,31)( 3,16,10,33)( 4,15, 9,34)( 5,17,11,35)( 6,18,12,36)(19,20)(21,22)(23,24)(25,38)(26,37)(27,40)(28,39)(29,42)(30,41)$
4C $4^{6},2^{9}$ $126$ $4$ $27$ $( 1,13)( 2,14)( 3,17)( 4,18)( 5,16)( 6,15)( 7,26,32,19)( 8,25,31,20)( 9,30,33,23)(10,29,34,24)(11,27,36,22)(12,28,35,21)(37,38)(39,42)(40,41)$
4D $4^{6},2^{8},1^{2}$ $126$ $4$ $26$ $( 1, 9,14,33)( 2,10,13,34)( 3, 7,15,32)( 4, 8,16,31)( 5,11,18,36)( 6,12,17,35)(19,27)(20,28)(21,25)(22,26)(23,30)(24,29)(37,40)(38,39)$
6A $6^{7}$ $2$ $6$ $35$ $( 1, 3, 6, 2, 4, 5)( 7, 9,11, 8,10,12)(13,16,18,14,15,17)(19,21,24,20,22,23)(25,27,30,26,28,29)(31,34,35,32,33,36)(37,39,42,38,40,41)$
6B $6^{7}$ $42$ $6$ $35$ $( 1, 5, 4, 2, 6, 3)( 7,29,10,26,11,27)( 8,30, 9,25,12,28)(13,17,15,14,18,16)(19,36,22,32,24,34)(20,35,21,31,23,33)(37,41,40,38,42,39)$
6C $6^{4},3^{6}$ $42$ $6$ $32$ $( 1, 4, 6)( 2, 3, 5)( 7,34,11,32,10,36)( 8,33,12,31, 9,35)(13,15,18)(14,16,17)(19,27,24,26,22,29)(20,28,23,25,21,30)(37,40,42)(38,39,41)$
6D $6^{6},2^{3}$ $56$ $6$ $33$ $( 1, 7,38, 2, 8,37)( 3, 9,40, 4,10,39)( 5,12,42, 6,11,41)(13,19,32,14,20,31)(15,22,34,16,21,33)(17,23,35,18,24,36)(25,26)(27,28)(29,30)$
6E $6^{7}$ $112$ $6$ $35$ $( 1,11,22, 2,12,21)( 3, 8,23, 4, 7,24)( 5, 9,20, 6,10,19)(13,29,40,14,30,39)(15,26,42,16,25,41)(17,28,38,18,27,37)(31,36,33,32,35,34)$
6F $6^{4},3^{4},2^{2},1^{2}$ $168$ $6$ $30$ $( 1,12,19, 6, 8,24)( 2,11,20, 5, 7,23)( 3,10,21)( 4, 9,22)(13,42,32,18,37,36)(14,41,31,17,38,35)(15,40,34)(16,39,33)(25,30)(26,29)$
6G $6^{6},2^{3}$ $168$ $6$ $33$ $( 1,42,14, 5,38,18)( 2,41,13, 6,37,17)( 3,39,15, 4,40,16)( 7,12)( 8,11)( 9,10)(19,36,26,23,31,30)(20,35,25,24,32,29)(21,33,28,22,34,27)$
7A1 $7^{6}$ $24$ $7$ $36$ $( 1,14,31,26, 8,38,19)( 2,13,32,25, 7,37,20)( 3,15,34,28,10,40,21)( 4,16,33,27, 9,39,22)( 5,18,36,30,11,42,23)( 6,17,35,29,12,41,24)$
7A-1 $7^{6}$ $24$ $7$ $36$ $( 1,19,38, 8,26,31,14)( 2,20,37, 7,25,32,13)( 3,21,40,10,28,34,15)( 4,22,39, 9,27,33,16)( 5,23,42,11,30,36,18)( 6,24,41,12,29,35,17)$
12A $12^{2},6^{2},3^{2}$ $84$ $12$ $36$ $( 1,17, 4,14, 6,16)( 2,18, 3,13, 5,15)( 7,23,34,25,11,21,32,30,10,20,36,28)( 8,24,33,26,12,22,31,29, 9,19,35,27)(37,42,40)(38,41,39)$
12B $12^{2},6^{3}$ $84$ $12$ $37$ $( 1,36, 9,13, 6,34, 8,18, 4,32,12,15)( 2,35,10,14, 5,33, 7,17, 3,31,11,16)(19,23,22,20,24,21)(25,41,28,38,30,39)(26,42,27,37,29,40)$
14A1 $14^{3}$ $24$ $14$ $39$ $( 1, 7,14,37,31,20,26, 2, 8,13,38,32,19,25)( 3, 9,15,39,34,22,28, 4,10,16,40,33,21,27)( 5,12,18,41,36,24,30, 6,11,17,42,35,23,29)$
14A-1 $14^{3}$ $24$ $14$ $39$ $( 1,25,19,32,38,13, 8, 2,26,20,31,37,14, 7)( 3,27,21,33,40,16,10, 4,28,22,34,39,15, 9)( 5,29,23,35,42,17,11, 6,30,24,36,41,18,12)$
14B1 $14^{2},7^{2}$ $72$ $14$ $38$ $( 1, 9,31,27,14,22,38, 4, 8,33,26,16,19,39)( 2,10,32,28,13,21,37, 3, 7,34,25,15,20,40)( 5,11,36,30,18,23,42)( 6,12,35,29,17,24,41)$
14B-1 $14^{2},7^{2}$ $72$ $14$ $38$ $( 1,39,19,16,26,33, 8, 4,38,22,14,27,31, 9)( 2,40,20,15,25,34, 7, 3,37,21,13,28,32,10)( 5,42,23,18,30,36,11)( 6,41,24,17,29,35,12)$
14C1 $14^{3}$ $72$ $14$ $39$ $( 1,42,31,30,19,18, 8, 5,38,36,26,23,14,11)( 2,41,32,29,20,17, 7, 6,37,35,25,24,13,12)( 3,39,34,27,21,16,10, 4,40,33,28,22,15, 9)$
14C-1 $14^{3}$ $72$ $14$ $39$ $( 1,11,14,23,26,36,38, 5, 8,18,19,30,31,42)( 2,12,13,24,25,35,37, 6, 7,17,20,29,32,41)( 3, 9,15,22,28,33,40, 4,10,16,21,27,34,39)$
21A1 $21^{2}$ $48$ $21$ $40$ $( 1,41,27,14,24, 9,31, 6,39,26,17,22, 8,35, 4,38,29,16,19,12,33)( 2,42,28,13,23,10,32, 5,40,25,18,21, 7,36, 3,37,30,15,20,11,34)$
21A-1 $21^{2}$ $48$ $21$ $40$ $( 1,33,12,19,16,29,38, 4,35, 8,22,17,26,39, 6,31, 9,24,14,27,41)( 2,34,11,20,15,30,37, 3,36, 7,21,18,25,40, 5,32,10,23,13,28,42)$
42A1 $42$ $48$ $42$ $41$ $( 1,21,41, 7,27,36,14, 3,24,37, 9,30,31,15, 6,20,39,11,26,34,17, 2,22,42, 8,28,35,13, 4,23,38,10,29,32,16, 5,19,40,12,25,33,18)$
42A-1 $42$ $48$ $42$ $41$ $( 1,18,33,25,12,40,19, 5,16,32,29,10,38,23, 4,13,35,28, 8,42,22, 2,17,34,26,11,39,20, 6,15,31,30, 9,37,24, 3,14,36,27, 7,41,21)$

Malle's constant $a(G)$:     $1/12$

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

36 x 36 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