Properties

Label 38T9
38T9 1 5 1->5 32 1->32 2 6 2->6 31 2->31 3 9 3->9 17 3->17 4 10 4->10 18 4->18 26 5->26 30 5->30 25 6->25 29 6->29 7 7->3 7->3 8 8->4 8->4 16 9->16 19 9->19 15 10->15 20 10->20 11 28 11->28 35 11->35 12 27 12->27 36 12->36 13 13->2 14 13->14 14->1 15->13 15->30 16->14 16->29 17->8 17->26 18->7 18->25 24 19->24 37 19->37 23 20->23 38 20->38 21 21->2 21->11 22 22->1 22->12 23->17 23->24 24->18 33 25->33 25->35 34 26->34 26->36 27->10 27->12 28->9 28->11 29->21 29->28 30->22 30->27 31->5 31->34 32->6 32->33 33->7 33->22 34->8 34->21 35->20 35->37 36->19 36->38 37->16 37->31 38->15 38->32
Degree $38$
Order $684$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2\times F_{19}$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(38, 9);
 
Copy content sage:G = TransitiveGroup(38, 9)
 
Copy content oscar:G = transitive_group(38, 9)
 
Copy content gap:G := TransitiveGroup(38, 9);
 

Group invariants

Abstract group:  $C_2\times F_{19}$
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:  $684=2^{2} \cdot 3^{2} \cdot 19$
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 $n$:  $38$
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 $t$:  $9$
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(38).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(38), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(38), G));
 
Generators:  $(1,32,6,25,33,22)(2,31,5,26,34,21)(3,9,19,24,18,7)(4,10,20,23,17,8)(11,35,37,16,29,28)(12,36,38,15,30,27)(13,14)$, $(1,5,30,22,12,27,10,15,13,2,6,29,21,11,28,9,16,14)(3,17,26,36,19,37,31,34,8,4,18,25,35,20,38,32,33,7)(23,24)$
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
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$9$:  $C_9$
$12$:  $C_6\times C_2$
$18$:  $C_{18}$ x 3
$36$:  36T2
$342$:  $F_{19}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: $F_{19}$

Low degree siblings

38T9

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{38}$ $1$ $1$ $0$ $()$
2A $2^{19}$ $1$ $2$ $19$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)$
2B $2^{18},1^{2}$ $19$ $2$ $18$ $( 3,38)( 4,37)( 5,36)( 6,35)( 7,34)( 8,33)( 9,32)(10,31)(11,29)(12,30)(13,28)(14,27)(15,25)(16,26)(17,24)(18,23)(19,21)(20,22)$
2C $2^{19}$ $19$ $2$ $19$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,38)(10,37)(11,35)(12,36)(13,34)(14,33)(15,31)(16,32)(17,30)(18,29)(19,27)(20,28)(21,25)(22,26)(23,24)$
3A1 $3^{12},1^{2}$ $19$ $3$ $24$ $( 3,23,16)( 4,24,15)( 5, 7,29)( 6, 8,30)( 9,14,20)(10,13,19)(11,36,34)(12,35,33)(17,25,37)(18,26,38)(21,31,28)(22,32,27)$
3A-1 $3^{12},1^{2}$ $19$ $3$ $24$ $( 3,16,23)( 4,15,24)( 5,29, 7)( 6,30, 8)( 9,20,14)(10,19,13)(11,34,36)(12,33,35)(17,37,25)(18,38,26)(21,28,31)(22,27,32)$
6A1 $6^{6},1^{2}$ $19$ $6$ $30$ $( 3,26,23,38,16,18)( 4,25,24,37,15,17)( 5,11, 7,36,29,34)( 6,12, 8,35,30,33)( 9,22,14,32,20,27)(10,21,13,31,19,28)$
6A-1 $6^{6},1^{2}$ $19$ $6$ $30$ $( 3,18,16,38,23,26)( 4,17,15,37,24,25)( 5,34,29,36, 7,11)( 6,33,30,35, 8,12)( 9,27,20,32,14,22)(10,28,19,31,13,21)$
6B1 $6^{6},2$ $19$ $6$ $31$ $( 1,11, 8, 2,12, 7)( 3,34,21, 4,33,22)( 5,18,36, 6,17,35)( 9,23,25,10,24,26)(13,29,16,14,30,15)(19,20)(27,31,37,28,32,38)$
6B-1 $6^{6},2$ $19$ $6$ $31$ $( 1, 7,12, 2, 8,11)( 3,22,33, 4,21,34)( 5,35,17, 6,36,18)( 9,26,24,10,25,23)(13,15,30,14,16,29)(19,20)(27,38,32,28,37,31)$
6C1 $6^{6},2$ $19$ $6$ $31$ $( 1,25,10, 7,21,37)( 2,26, 9, 8,22,38)( 3,11,31, 5,35,15)( 4,12,32, 6,36,16)(13,17,28,34,30,20)(14,18,27,33,29,19)(23,24)$
6C-1 $6^{6},2$ $19$ $6$ $31$ $( 1,37,21, 7,10,25)( 2,38,22, 8, 9,26)( 3,15,35, 5,31,11)( 4,16,36, 6,32,12)(13,20,30,34,28,17)(14,19,29,33,27,18)(23,24)$
9A1 $9^{4},1^{2}$ $19$ $9$ $32$ $( 3,33,19,23,12,10,16,35,13)( 4,34,20,24,11, 9,15,36,14)( 5,27,37, 7,22,17,29,32,25)( 6,28,38, 8,21,18,30,31,26)$
9A-1 $9^{4},1^{2}$ $19$ $9$ $32$ $( 3,13,35,16,10,12,23,19,33)( 4,14,36,15, 9,11,24,20,34)( 5,25,32,29,17,22, 7,37,27)( 6,26,31,30,18,21, 8,38,28)$
9A2 $9^{4},1^{2}$ $19$ $9$ $32$ $( 3,19,12,16,13,33,23,10,35)( 4,20,11,15,14,34,24, 9,36)( 5,37,22,29,25,27, 7,17,32)( 6,38,21,30,26,28, 8,18,31)$
9A-2 $9^{4},1^{2}$ $19$ $9$ $32$ $( 3,35,10,23,33,13,16,12,19)( 4,36, 9,24,34,14,15,11,20)( 5,32,17, 7,27,25,29,22,37)( 6,31,18, 8,28,26,30,21,38)$
9A4 $9^{4},1^{2}$ $19$ $9$ $32$ $( 3,12,13,23,35,19,16,33,10)( 4,11,14,24,36,20,15,34, 9)( 5,22,25, 7,32,37,29,27,17)( 6,21,26, 8,31,38,30,28,18)$
9A-4 $9^{4},1^{2}$ $19$ $9$ $32$ $( 3,10,33,16,19,35,23,13,12)( 4, 9,34,15,20,36,24,14,11)( 5,17,27,29,37,32, 7,25,22)( 6,18,28,30,38,31, 8,26,21)$
18A1 $18^{2},1^{2}$ $19$ $18$ $34$ $( 3,31,33,26,19, 6,23,28,12,38,10, 8,16,21,35,18,13,30)( 4,32,34,25,20, 5,24,27,11,37, 9, 7,15,22,36,17,14,29)$
18A-1 $18^{2},1^{2}$ $19$ $18$ $34$ $( 3,30,13,18,35,21,16, 8,10,38,12,28,23, 6,19,26,33,31)( 4,29,14,17,36,22,15, 7, 9,37,11,27,24, 5,20,25,34,32)$
18A5 $18^{2},1^{2}$ $19$ $18$ $34$ $( 3, 6,10,18,33,28,16,30,19,38,35,31,23, 8,13,26,12,21)( 4, 5, 9,17,34,27,15,29,20,37,36,32,24, 7,14,25,11,22)$
18A-5 $18^{2},1^{2}$ $19$ $18$ $34$ $( 3,21,12,26,13, 8,23,31,35,38,19,30,16,28,33,18,10, 6)( 4,22,11,25,14, 7,24,32,36,37,20,29,15,27,34,17, 9, 5)$
18A7 $18^{2},1^{2}$ $19$ $18$ $34$ $( 3,28,35,26,10,30,23,21,33,38,13, 6,16,31,12,18,19, 8)( 4,27,36,25, 9,29,24,22,34,37,14, 5,15,32,11,17,20, 7)$
18A-7 $18^{2},1^{2}$ $19$ $18$ $34$ $( 3, 8,19,18,12,31,16, 6,13,38,33,21,23,30,10,26,35,28)( 4, 7,20,17,11,32,15, 5,14,37,34,22,24,29, 9,25,36,27)$
18B1 $18^{2},2$ $19$ $18$ $35$ $( 1,36,10,11, 6,24, 8,17,26, 2,35, 9,12, 5,23, 7,18,25)( 3,29,28,34,16,32,21,14,38, 4,30,27,33,15,31,22,13,37)(19,20)$
18B-1 $18^{2},2$ $19$ $18$ $35$ $( 1,36,12,20,30,14,31,25,28, 2,35,11,19,29,13,32,26,27)( 3, 9, 8,34,38,24,16, 5,21, 4,10, 7,33,37,23,15, 6,22)(17,18)$
18B5 $18^{2},2$ $19$ $18$ $35$ $( 1,36,19,32, 3, 5,13, 7,21, 2,35,20,31, 4, 6,14, 8,22)( 9,30,34,12,37,28,25,18,24,10,29,33,11,38,27,26,17,23)(15,16)$
18B-5 $18^{2},2$ $19$ $18$ $35$ $( 1,36,16,29,23,32,33, 5,18, 2,35,15,30,24,31,34, 6,17)( 3, 7,28,14,19,11,10,37,26, 4, 8,27,13,20,12, 9,38,25)(21,22)$
18B7 $18^{2},2$ $19$ $18$ $35$ $( 1,36, 6,27,21,34,10,20,38, 2,35, 5,28,22,33, 9,19,37)( 3,32,13,11,16, 7,23,29,18, 4,31,14,12,15, 8,24,30,17)(25,26)$
18B-7 $18^{2},2$ $19$ $18$ $35$ $( 1,36,38,17,28, 4,16, 9,31, 2,35,37,18,27, 3,15,10,32)( 5,33,20, 8,14,30,22,26,24, 6,34,19, 7,13,29,21,25,23)(11,12)$
18C1 $18^{2},2$ $19$ $18$ $35$ $( 1,36,13,25,16,17,10, 4,28, 7,12,34,21,32,30,37, 6,20)( 2,35,14,26,15,18, 9, 3,27, 8,11,33,22,31,29,38, 5,19)(23,24)$
18C-1 $18^{2},2$ $19$ $18$ $35$ $( 1,36,18,32,38, 7, 6,15, 3,25,30, 9,33,27,19,22,12,24)( 2,35,17,31,37, 8, 5,16, 4,26,29,10,34,28,20,21,11,23)(13,14)$
18C5 $18^{2},2$ $19$ $18$ $35$ $( 1,36,28,11,18,29,16,25, 8, 9,13,22,38,32,19,34,23, 4)( 2,35,27,12,17,30,15,26, 7,10,14,21,37,31,20,33,24, 3)( 5, 6)$
18C-5 $18^{2},2$ $19$ $18$ $35$ $( 1,36,33,14, 3,17, 6,37,16,24,28,29,12,22, 8,20,26, 9)( 2,35,34,13, 4,18, 5,38,15,23,27,30,11,21, 7,19,25,10)(31,32)$
18C7 $18^{2},2$ $19$ $18$ $35$ $( 1,36,21,29,19, 4,23,17,16,27,31, 7,38, 9,26, 5,12,14)( 2,35,22,30,20, 3,24,18,15,28,32, 8,37,10,25, 6,11,13)(33,34)$
18C-7 $18^{2},2$ $19$ $18$ $35$ $( 1,36,23,25,31,11,28,37,30, 5,10,22,19,14,33,17, 8,15)( 2,35,24,26,32,12,27,38,29, 6, 9,21,20,13,34,18, 7,16)( 3, 4)$
19A $19^{2}$ $18$ $19$ $36$ $( 1,31,23,16, 8,38,30,21,13, 6,35,28,19,12, 3,33,26,18,10)( 2,32,24,15, 7,37,29,22,14, 5,36,27,20,11, 4,34,25,17, 9)$
38A $38$ $18$ $38$ $37$ $( 1,36,31,27,23,20,16,11, 8, 4,38,34,30,25,21,17,13, 9, 6, 2,35,32,28,24,19,15,12, 7, 3,37,33,29,26,22,18,14,10, 5)$

Malle's constant $a(G)$:     $1/18$

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

38 x 38 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