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Properties

Label 38T28
38T28 1 24 1->24 38 1->38 2 26 2->26 35 2->35 3 28 3->28 32 3->32 4 29 4->29 30 4->30 5 5->26 5->32 6 23 6->23 34 6->34 7 20 7->20 36 7->36 8 8->36 8->38 9 21 9->21 33 9->33 10 10->23 10->30 11 25 11->25 27 11->27 12 12->24 12->27 13 13->21 13->29 14 31 14->31 37 14->37 15 15->33 15->34 16 16->31 16->35 17 17->28 17->37 18 18->20 18->25 19 22 19->22 19->22 20->10 20->19 21->14 21->15 22->11 22->18 23->3 23->7 24->3 24->7 25->11 25->18 26->14 26->15 27->10 27->19 28->4 28->6 29->2 29->8 30->12 30->17 31->13 32->1 32->9 33->5 33->5 34->1 34->9 35->13 36->12 36->17 37->2 37->8 38->4 38->6
Degree $38$
Order $12996$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_{19}^2:(C_6\times S_3)$

Related objects

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(38, 28);
 
Copy content sage:G = TransitiveGroup(38, 28)
 
Copy content oscar:G = transitive_group(38, 28)
 
Copy content gap:G := TransitiveGroup(38, 28);
 

Group invariants

Abstract group:  $C_{19}^2:(C_6\times S_3)$
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:  $12996=2^{2} \cdot 3^{2} \cdot 19^{2}$
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 $t$:  $28$
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);
 
$\card{\Aut(F/K)}$:  $1$
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:  $(1,24,3,28,6,34)(2,26,14,31,13,29)(4,30,17,37,8,38)(5,32,9,21,15,33)(7,36,12,27,10,23)(11,25,18,20,19,22)(16,35)$, $(1,38,6,23,3,32)(2,35,13,21,14,37)(4,29,8,36,17,28)(5,26,15,34,9,33)(7,20,10,30,12,24)(11,27,19,22,18,25)(16,31)$
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
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$36$:  $C_6\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: 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 $1^{38}$ $1$ $1$ $0$ $()$
2A $2^{19}$ $57$ $2$ $19$ $( 1,25)( 2,27)( 3,29)( 4,31)( 5,33)( 6,35)( 7,37)( 8,20)( 9,22)(10,24)(11,26)(12,28)(13,30)(14,32)(15,34)(16,36)(17,38)(18,21)(19,23)$
2B $2^{19}$ $57$ $2$ $19$ $( 1,28)( 2,25)( 3,22)( 4,38)( 5,35)( 6,32)( 7,29)( 8,26)( 9,23)(10,20)(11,36)(12,33)(13,30)(14,27)(15,24)(16,21)(17,37)(18,34)(19,31)$
2C $2^{18},1^{2}$ $361$ $2$ $18$ $( 1, 6)( 2, 5)( 3, 4)( 7,19)( 8,18)( 9,17)(10,16)(11,15)(12,14)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(37,38)$
3A1 $3^{6},1^{20}$ $38$ $3$ $12$ $(20,22,25)(21,33,32)(23,36,27)(24,28,34)(26,31,29)(30,37,38)$
3A-1 $3^{6},1^{20}$ $38$ $3$ $12$ $(20,25,22)(21,32,33)(23,27,36)(24,34,28)(26,29,31)(30,38,37)$
3B1 $3^{12},1^{2}$ $361$ $3$ $24$ $( 1,14,10)( 3, 9,13)( 4,16, 5)( 6,11, 8)( 7,18,19)(12,15,17)(20,28,27)(21,35,38)(22,23,30)(24,37,33)(26,32,36)(29,34,31)$
3B-1 $3^{12},1^{2}$ $361$ $3$ $24$ $( 1,10,14)( 3,13, 9)( 4, 5,16)( 6, 8,11)( 7,19,18)(12,17,15)(20,27,28)(21,38,35)(22,30,23)(24,33,37)(26,36,32)(29,31,34)$
3C $3^{12},1^{2}$ $722$ $3$ $24$ $( 1, 5,14)( 2,12, 6)( 3,19,17)( 4, 7, 9)( 8,16,15)(10,11,18)(20,35,29)(21,27,36)(22,38,24)(23,30,31)(25,33,26)(32,34,37)$
6A1 $6^{6},1^{2}$ $361$ $6$ $30$ $( 1,11,17,13, 3,16)( 2, 4, 9,12,10, 5)( 6,14,15, 8,19,18)(20,36,38,24,27,25)(21,29,30,23,34,33)(26,32,28,37,31,35)$
6A-1 $6^{6},1^{2}$ $361$ $6$ $30$ $( 1,16, 3,13,17,11)( 2, 5,10,12, 9, 4)( 6,18,19, 8,15,14)(20,25,27,24,38,36)(21,33,34,23,30,29)(26,35,31,37,28,32)$
6B $6^{6},1^{2}$ $722$ $6$ $30$ $( 1,12, 5, 6,14, 2)( 3, 9,19, 4,17, 7)( 8,11,16,18,15,10)(20,27,35,36,29,21)(22,32,38,34,24,37)(23,25,30,33,31,26)$
6C1 $6^{3},2^{9},1^{2}$ $722$ $6$ $24$ $( 1,18)( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(20,26,22,31,25,29)(21,38,33,30,32,37)(23,24,36,28,27,34)$
6C-1 $6^{3},2^{9},1^{2}$ $722$ $6$ $24$ $( 1,18)( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(20,29,25,31,22,26)(21,37,32,30,33,38)(23,34,27,28,36,24)$
6D1 $6^{6},2$ $1083$ $6$ $31$ $( 1,20,14,28,10,27)( 2,25)( 3,30, 9,22,13,23)( 4,35,16,38, 5,21)( 6,26,11,32, 8,36)( 7,31,18,29,19,34)(12,37,15,33,17,24)$
6D-1 $6^{6},2$ $1083$ $6$ $31$ $( 1,27,10,28,14,20)( 2,25)( 3,23,13,22, 9,30)( 4,21, 5,38,16,35)( 6,36, 8,32,11,26)( 7,34,19,29,18,31)(12,24,17,33,15,37)$
6E1 $6^{6},2$ $1083$ $6$ $31$ $( 1,24,10,32,16,31)( 2,27,17,34, 8,26)( 3,30, 5,36,19,21)( 4,33,12,38,11,35)( 6,20, 7,23,14,25)( 9,29)(13,22,18,37,15,28)$
6E-1 $6^{6},2$ $1083$ $6$ $31$ $( 1,31,16,32,10,24)( 2,26, 8,34,17,27)( 3,21,19,36, 5,30)( 4,35,11,38,12,33)( 6,25,14,23, 7,20)( 9,29)(13,28,15,37,18,22)$
19A1 $19,1^{19}$ $12$ $19$ $18$ $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)$
19A2 $19,1^{19}$ $12$ $19$ $18$ $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)$
19A4 $19,1^{19}$ $12$ $19$ $18$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)$
19B1 $19^{2}$ $18$ $19$ $36$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,26,32,38,25,31,37,24,30,36,23,29,35,22,28,34,21,27,33)$
19B2 $19^{2}$ $18$ $19$ $36$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,32,25,37,30,23,35,28,21,33,26,38,31,24,36,29,22,34,27)$
19B4 $19^{2}$ $18$ $19$ $36$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,25,30,35,21,26,31,36,22,27,32,37,23,28,33,38,24,29,34)$
19C1 $19^{2}$ $18$ $19$ $36$ $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$
19C2 $19^{2}$ $18$ $19$ $36$ $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,28,36,25,33,22,30,38,27,35,24,32,21,29,37,26,34,23,31)$
19C4 $19^{2}$ $18$ $19$ $36$ $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,36,33,30,27,24,21,37,34,31,28,25,22,38,35,32,29,26,23)$
19D1 $19^{2}$ $36$ $19$ $36$ $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)(20,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21)$
19D2 $19^{2}$ $36$ $19$ $36$ $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,37,35,33,31,29,27,25,23,21,38,36,34,32,30,28,26,24,22)$
19D4 $19^{2}$ $36$ $19$ $36$ $( 1,17,14,11, 8, 5, 2,18,15,12, 9, 6, 3,19,16,13,10, 7, 4)(20,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21)$
19E1 $19^{2}$ $36$ $19$ $36$ $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21)$
19E2 $19^{2}$ $36$ $19$ $36$ $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21)$
19E4 $19^{2}$ $36$ $19$ $36$ $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,35,31,27,23,38,34,30,26,22,37,33,29,25,21,36,32,28,24)$
38A1 $38$ $342$ $38$ $37$ $( 1,28, 4,34, 7,21,10,27,13,33,16,20,19,26, 3,32, 6,38, 9,25,12,31,15,37,18,24, 2,30, 5,36, 8,23,11,29,14,35,17,22)$
38A3 $38$ $342$ $38$ $37$ $( 1,34,10,33,19,32, 9,31,18,30, 8,29,17,28, 7,27,16,26, 6,25,15,24, 5,23,14,22, 4,21,13,20, 3,38,12,37, 2,36,11,35)$
38A9 $38$ $342$ $38$ $37$ $( 1,33, 9,30,17,27, 6,24,14,21, 3,37,11,34,19,31, 8,28,16,25, 5,22,13,38, 2,35,10,32,18,29, 7,26,15,23, 4,20,12,36)$
38B1 $38$ $342$ $38$ $37$ $( 1,29,18,33,16,37,14,22,12,26,10,30, 8,34, 6,38, 4,23, 2,27,19,31,17,35,15,20,13,24,11,28, 9,32, 7,36, 5,21, 3,25)$
38B3 $38$ $342$ $38$ $37$ $( 1,33,14,26, 8,38, 2,31,15,24, 9,36, 3,29,16,22,10,34, 4,27,17,20,11,32, 5,25,18,37,12,30, 6,23,19,35,13,28, 7,21)$
38B9 $38$ $342$ $38$ $37$ $( 1,26, 2,24, 3,22, 4,20, 5,37, 6,35, 7,33, 8,31, 9,29,10,27,11,25,12,23,13,21,14,38,15,36,16,34,17,32,18,30,19,28)$
57A1 $19,3^{6},1$ $228$ $57$ $30$ $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,33,29)(22,28,32)(23,35,24)(25,30,27)(26,37,38)(31,34,36)$
57A-1 $19,3^{6},1$ $228$ $57$ $30$ $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,29,33)(22,32,28)(23,24,35)(25,27,30)(26,38,37)(31,36,34)$
57A2 $19,3^{6},1$ $228$ $57$ $30$ $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,29,33)(22,32,28)(23,24,35)(25,27,30)(26,38,37)(31,36,34)$
57A-2 $19,3^{6},1$ $228$ $57$ $30$ $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,33,29)(22,28,32)(23,35,24)(25,30,27)(26,37,38)(31,34,36)$
57A4 $19,3^{6},1$ $228$ $57$ $30$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,33,29)(22,28,32)(23,35,24)(25,30,27)(26,37,38)(31,34,36)$
57A-4 $19,3^{6},1$ $228$ $57$ $30$ $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,29,33)(22,32,28)(23,24,35)(25,27,30)(26,38,37)(31,36,34)$

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

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