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Properties

Label 42T50
42T50 1 5 1->5 40 1->40 2 4 2->4 42 2->42 3 6 3->6 41 3->41 4->1 4->3 5->2 5->3 6->1 6->2 7 7->6 7->40 8 8->5 8->42 9 9->4 9->41 10 10->7 39 10->39 11 11->9 38 11->38 12 12->8 37 12->37 13 13->11 35 13->35 14 14->10 36 14->36 15 15->12 34 15->34 16 16->15 31 16->31 17 17->13 33 17->33 18 18->14 32 18->32 19 19->17 30 19->30 20 20->16 29 20->29 21 21->18 28 21->28 22 22->19 27 22->27 23 23->21 26 23->26 24 24->20 25 24->25 25->23 25->24 26->22 26->23 27->22 27->24 28->19 28->26 29->21 29->25 30->20 30->27 31->18 31->30 32->17 32->29 33->16 33->28 34->14 34->31 35->15 35->33 36->13 36->32 37->10 37->35 38->12 38->36 39->11 39->34 40->9 40->39 41->8 41->38 42->7 42->37
Degree $42$
Order $252$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_{21}:D_6$

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, 50);
 
Copy content sage:G = TransitiveGroup(42, 50)
 
Copy content oscar:G = transitive_group(42, 50)
 
Copy content gap:G := TransitiveGroup(42, 50);
 

Group invariants

Abstract group:  $C_{21}:D_6$
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:  $252=2^{2} \cdot 3^{2} \cdot 7$
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:  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 $t$:  $50$
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);
 
$\card{\Aut(F/K)}$:  $3$
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:  $(1,40,39,34,31,30,27,22,19,17,13,11,9,4,3,41,38,36,32,29,25,24,20,16,15,12,8,5,2,42,37,35,33,28,26,23,21,18,14,10,7,6)$, $(1,5,3,6,2,4)(7,40,9,41,8,42)(10,39,11,38,12,37)(13,35,15,34,14,36)(16,31,18,32,17,33)(19,30,20,29,21,28)(22,27,24,25,23,26)$
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
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$14$:  $D_{7}$
$18$:  $S_3\times C_3$
$28$:  $D_{14}$
$36$:  $C_6\times S_3$
$42$:  21T3
$84$:  21T8, 42T9

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $S_3\times C_3$

Degree 7: $D_{7}$

Degree 14: $D_{14}$

Degree 21: None

Low degree siblings

There are no siblings 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 $2^{21}$ $3$ $2$ $21$ $( 1,23)( 2,24)( 3,22)( 4,25)( 5,26)( 6,27)( 7,28)( 8,29)( 9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)$
2B $2^{18},1^{6}$ $7$ $2$ $18$ $( 4,40)( 5,41)( 6,42)( 7,38)( 8,39)( 9,37)(10,34)(11,36)(12,35)(13,31)(14,33)(15,32)(16,30)(17,28)(18,29)(19,25)(20,26)(21,27)$
2C $2^{21}$ $21$ $2$ $21$ $( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,36)( 8,34)( 9,35)(10,33)(11,31)(12,32)(13,30)(14,28)(15,29)(16,26)(17,25)(18,27)(19,24)(20,23)(21,22)$
3A1 $3^{14}$ $1$ $3$ $28$ $( 1, 3, 2)( 4, 5, 6)( 7, 9, 8)(10,11,12)(13,15,14)(16,18,17)(19,20,21)(22,24,23)(25,26,27)(28,30,29)(31,32,33)(34,36,35)(37,39,38)(40,41,42)$
3A-1 $3^{14}$ $1$ $3$ $28$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,14,15)(16,17,18)(19,21,20)(22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,35,36)(37,38,39)(40,42,41)$
3B $3^{14}$ $2$ $3$ $28$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,38,39)(40,41,42)$
3C1 $3^{7},1^{21}$ $2$ $3$ $14$ $( 4, 6, 5)(10,12,11)(16,17,18)(22,23,24)(28,29,30)(34,35,36)(40,42,41)$
3C-1 $3^{7},1^{21}$ $2$ $3$ $14$ $( 4, 5, 6)(10,11,12)(16,18,17)(22,24,23)(28,30,29)(34,36,35)(40,41,42)$
6A1 $6^{7}$ $3$ $6$ $35$ $( 1,22, 2,23, 3,24)( 4,26, 6,25, 5,27)( 7,30, 8,28, 9,29)(10,32,12,31,11,33)(13,36,14,34,15,35)(16,39,17,37,18,38)(19,41,21,40,20,42)$
6A-1 $6^{7}$ $3$ $6$ $35$ $( 1,24, 3,23, 2,22)( 4,27, 5,25, 6,26)( 7,29, 9,28, 8,30)(10,33,11,31,12,32)(13,35,15,34,14,36)(16,38,18,37,17,39)(19,42,20,40,21,41)$
6B1 $6^{6},3^{2}$ $7$ $6$ $34$ $( 1, 2, 3)( 4,42, 5,40, 6,41)( 7,39, 9,38, 8,37)(10,35,11,34,12,36)(13,33,15,31,14,32)(16,28,18,30,17,29)(19,27,20,25,21,26)(22,23,24)$
6B-1 $6^{6},3^{2}$ $7$ $6$ $34$ $( 1, 3, 2)( 4,41, 6,40, 5,42)( 7,37, 8,38, 9,39)(10,36,12,34,11,35)(13,32,14,31,15,33)(16,29,17,30,18,28)(19,26,21,25,20,27)(22,24,23)$
6C $6^{6},3^{2}$ $14$ $6$ $34$ $( 1, 8, 2, 9, 3, 7)( 4, 6, 5)(10,40,11,41,12,42)(13,37,14,38,15,39)(16,34,18,36,17,35)(19,31,21,33,20,32)(22,30,24,29,23,28)(25,26,27)$
6D1 $6^{3},3,2^{9},1^{3}$ $14$ $6$ $26$ $( 1, 9)( 2, 7)( 3, 8)( 4, 5, 6)(10,42,12,41,11,40)(13,38)(14,39)(15,37)(16,35,17,36,18,34)(19,33)(20,31)(21,32)(22,28,23,29,24,30)$
6D-1 $6^{3},3,2^{9},1^{3}$ $14$ $6$ $26$ $( 1, 7, 3, 9, 2, 8)(10,41)(11,42)(12,40)(13,39,15,38,14,37)(16,36)(17,34)(18,35)(19,32,20,33,21,31)(22,29)(23,30)(24,28)(25,27,26)$
6E1 $6^{7}$ $21$ $6$ $35$ $( 1,41, 2,40, 3,42)( 4,39, 6,37, 5,38)( 7,35, 8,36, 9,34)(10,31,12,33,11,32)(13,29,14,30,15,28)(16,27,17,26,18,25)(19,23,21,24,20,22)$
6E-1 $6^{7}$ $21$ $6$ $35$ $( 1,42, 3,40, 2,41)( 4,38, 5,37, 6,39)( 7,34, 9,36, 8,35)(10,32,11,33,12,31)(13,28,15,30,14,29)(16,25,18,26,17,27)(19,22,20,24,21,23)$
7A1 $7^{6}$ $2$ $7$ $36$ $( 1,21,37,15,32, 9,27)( 2,20,38,13,31, 7,26)( 3,19,39,14,33, 8,25)( 4,22,40,18,35,12,29)( 5,24,41,17,34,10,28)( 6,23,42,16,36,11,30)$
7A2 $7^{6}$ $2$ $7$ $36$ $( 1,37,32,27,21,15, 9)( 2,38,31,26,20,13, 7)( 3,39,33,25,19,14, 8)( 4,40,35,29,22,18,12)( 5,41,34,28,24,17,10)( 6,42,36,30,23,16,11)$
7A3 $7^{6}$ $2$ $7$ $36$ $( 1,15,27,37, 9,21,32)( 2,13,26,38, 7,20,31)( 3,14,25,39, 8,19,33)( 4,18,29,40,12,22,35)( 5,17,28,41,10,24,34)( 6,16,30,42,11,23,36)$
14A1 $14^{3}$ $6$ $14$ $39$ $( 1,36,27,16, 9,42,32,23,15, 6,37,30,21,11)( 2,34,26,17, 7,41,31,24,13, 5,38,28,20,10)( 3,35,25,18, 8,40,33,22,14, 4,39,29,19,12)$
14A3 $14^{3}$ $6$ $14$ $39$ $( 1,16,32, 6,21,36, 9,23,37,11,27,42,15,30)( 2,17,31, 5,20,34, 7,24,38,10,26,41,13,28)( 3,18,33, 4,19,35, 8,22,39,12,25,40,14,29)$
14A5 $14^{3}$ $6$ $14$ $39$ $( 1,42,37,36,32,30,27,23,21,16,15,11, 9, 6)( 2,41,38,34,31,28,26,24,20,17,13,10, 7, 5)( 3,40,39,35,33,29,25,22,19,18,14,12, 8, 4)$
21A1 $21^{2}$ $2$ $21$ $40$ $( 1,38,33,27,20,14, 9, 2,39,32,26,19,15, 7, 3,37,31,25,21,13, 8)( 4,42,34,29,23,17,12, 6,41,35,30,24,18,11, 5,40,36,28,22,16,10)$
21A-1 $21^{2}$ $2$ $21$ $40$ $( 1, 8,13,21,25,31,37, 3, 7,15,19,26,32,39, 2, 9,14,20,27,33,38)( 4,10,16,22,28,36,40, 5,11,18,24,30,35,41, 6,12,17,23,29,34,42)$
21A2 $21^{2}$ $2$ $21$ $40$ $( 1,33,20, 9,39,26,15, 3,31,21, 8,38,27,14, 2,32,19, 7,37,25,13)( 4,34,23,12,41,30,18, 5,36,22,10,42,29,17, 6,35,24,11,40,28,16)$
21A-2 $21^{2}$ $2$ $21$ $40$ $( 1,13,25,37, 7,19,32, 2,14,27,38, 8,21,31, 3,15,26,39, 9,20,33)( 4,16,28,40,11,24,35, 6,17,29,42,10,22,36, 5,18,30,41,12,23,34)$
21A4 $21^{2}$ $2$ $21$ $40$ $( 1,20,39,15,31, 8,27, 2,19,37,13,33, 9,26, 3,21,38,14,32, 7,25)( 4,23,41,18,36,10,29, 6,24,40,16,34,12,30, 5,22,42,17,35,11,28)$
21A-4 $21^{2}$ $2$ $21$ $40$ $( 1,25, 7,32,14,38,21, 3,26, 9,33,13,37,19, 2,27, 8,31,15,39,20)( 4,28,11,35,17,42,22, 5,30,12,34,16,40,24, 6,29,10,36,18,41,23)$
21B1 $21^{2}$ $4$ $21$ $40$ $( 1, 8,13,21,25,31,37, 3, 7,15,19,26,32,39, 2, 9,14,20,27,33,38)( 4,11,17,22,30,34,40, 6,10,18,23,28,35,42, 5,12,16,24,29,36,41)$
21B2 $21^{2}$ $4$ $21$ $40$ $( 1,13,25,37, 7,19,32, 2,14,27,38, 8,21,31, 3,15,26,39, 9,20,33)( 4,17,30,40,10,23,35, 5,16,29,41,11,22,34, 6,18,28,42,12,24,36)$
21B4 $21^{2}$ $4$ $21$ $40$ $( 1,25, 7,32,14,38,21, 3,26, 9,33,13,37,19, 2,27, 8,31,15,39,20)( 4,30,10,35,16,41,22, 6,28,12,36,17,40,23, 5,29,11,34,18,42,24)$
21C1 $21,7^{3}$ $4$ $21$ $38$ $( 1, 7,14,21,26,33,37, 2, 8,15,20,25,32,38, 3, 9,13,19,27,31,39)( 4,12,18,22,29,35,40)( 5,10,17,24,28,34,41)( 6,11,16,23,30,36,42)$
21C-1 $21,7^{3}$ $4$ $21$ $38$ $( 1, 9,15,21,27,32,37)( 2, 7,13,20,26,31,38)( 3, 8,14,19,25,33,39)( 4,10,16,22,28,36,40, 5,11,18,24,30,35,41, 6,12,17,23,29,34,42)$
21C2 $21,7^{3}$ $4$ $21$ $38$ $( 1,14,26,37, 8,20,32, 3,13,27,39, 7,21,33, 2,15,25,38, 9,19,31)( 4,18,29,40,12,22,35)( 5,17,28,41,10,24,34)( 6,16,30,42,11,23,36)$
21C-2 $21,7^{3}$ $4$ $21$ $38$ $( 1,15,27,37, 9,21,32)( 2,13,26,38, 7,20,31)( 3,14,25,39, 8,19,33)( 4,16,28,40,11,24,35, 6,17,29,42,10,22,36, 5,18,30,41,12,23,34)$
21C4 $21,7^{3}$ $4$ $21$ $38$ $( 1,26, 8,32,13,39,21, 2,25, 9,31,14,37,20, 3,27, 7,33,15,38,19)( 4,29,12,35,18,40,22)( 5,28,10,34,17,41,24)( 6,30,11,36,16,42,23)$
21C-4 $21,7^{3}$ $4$ $21$ $38$ $( 1,27, 9,32,15,37,21)( 2,26, 7,31,13,38,20)( 3,25, 8,33,14,39,19)( 4,28,11,35,17,42,22, 5,30,12,34,16,40,24, 6,29,10,36,18,41,23)$
42A1 $42$ $6$ $42$ $41$ $( 1,40,38,36,33,28,27,22,20,16,14,10, 9, 4, 2,42,39,34,32,29,26,23,19,17,15,12, 7, 6, 3,41,37,35,31,30,25,24,21,18,13,11, 8, 5)$
42A-1 $42$ $6$ $42$ $41$ $( 1,41,39,36,31,29,27,24,19,16,13,12, 9, 5, 3,42,38,35,32,28,25,23,20,18,15,10, 8, 6, 2,40,37,34,33,30,26,22,21,17,14,11, 7, 4)$
42A5 $42$ $6$ $42$ $41$ $( 1,29,14,41,26,11,37,22, 8,34,20, 6,32,18, 3,28,13,42,27,12,39,24, 7,36,21, 4,33,17, 2,30,15,40,25,10,38,23, 9,35,19, 5,31,16)$
42A-5 $42$ $6$ $42$ $41$ $( 1,30,13,41,25,12,37,23, 7,34,19, 4,32,16, 2,28,14,40,27,11,38,24, 8,35,21, 6,31,17, 3,29,15,42,26,10,39,22, 9,36,20, 5,33,18)$
42A11 $42$ $6$ $42$ $41$ $( 1,36,25,18, 7,41,32,23,14, 4,38,28,21,11, 3,35,26,17, 9,42,33,22,13, 5,37,30,19,12, 2,34,27,16, 8,40,31,24,15, 6,39,29,20,10)$
42A-11 $42$ $6$ $42$ $41$ $( 1,34,26,18, 8,42,32,24,13, 4,39,30,21,10, 2,35,25,16, 9,41,31,22,14, 6,37,28,20,12, 3,36,27,17, 7,40,33,23,15, 5,38,29,19,11)$

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

45 x 45 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