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Properties

Label 38T30
38T30 1 6 1->6 29 1->29 2 5 2->5 31 2->31 3 4 3->4 33 3->33 35 4->35 37 5->37 20 6->20 7 19 7->19 22 7->22 8 18 8->18 24 8->24 9 17 9->17 26 9->26 10 16 10->16 28 10->28 11 15 11->15 30 11->30 12 14 12->14 32 12->32 13 34 13->34 36 14->36 38 15->38 21 16->21 23 17->23 25 18->25 27 19->27 20->13 20->30 21->2 21->29 22->10 22->28 23->18 23->27 24->7 24->26 25->15 26->4 27->12 28->1 29->9 30->17 31->6 31->38 32->14 32->37 33->36 34->11 34->35 35->19 36->8 37->16 38->5
Degree $38$
Order $12996$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{19}:F_{19}$

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Copy content magma:G := TransitiveGroup(38, 30);
 

Group invariants

Abstract group:  $D_{19}:F_{19}$
Copy content magma:IdentifyGroup(G);
 
Order:  $12996=2^{2} \cdot 3^{2} \cdot 19^{2}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $38$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $30$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,6)(2,5)(3,4)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(20,30)(21,29)(22,28)(23,27)(24,26)(31,38)(32,37)(33,36)(34,35)$, $(1,29,9,26,4,35,19,27,12,32,14,36,8,24,7,22,10,28)(2,31,6,20,13,34,11,30,17,23,18,25,15,38,5,37,16,21)(3,33)$
Copy content magma:Generators(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
$342$:  $F_{19}$ x 2
$684$:  38T9 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: None

Low degree siblings

38T30 x 8

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 $2^{19}$ $19$ $2$ $19$ $( 1,38)( 2,28)( 3,37)( 4,27)( 5,36)( 6,26)( 7,35)( 8,25)( 9,34)(10,24)(11,33)(12,23)(13,32)(14,22)(15,31)(16,21)(17,30)(18,20)(19,29)$
2B $2^{19}$ $19$ $2$ $19$ $( 1,25)( 2,35)( 3,26)( 4,36)( 5,27)( 6,37)( 7,28)( 8,38)( 9,29)(10,20)(11,30)(12,21)(13,31)(14,22)(15,32)(16,23)(17,33)(18,24)(19,34)$
2C $2^{18},1^{2}$ $361$ $2$ $18$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,19)(10,18)(11,17)(12,16)(13,15)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(36,38)$
3A1 $3^{12},1^{2}$ $361$ $3$ $24$ $( 1,18, 4)( 2, 6,15)( 3,13, 7)( 5, 8,10)( 9,17,16)(11,12,19)(20,32,21)(22,27,24)(23,34,35)(25,29,38)(26,36,30)(28,31,33)$
3A-1 $3^{12},1^{2}$ $361$ $3$ $24$ $( 1, 4,18)( 2,15, 6)( 3, 7,13)( 5,10, 8)( 9,16,17)(11,19,12)(20,21,32)(22,24,27)(23,35,34)(25,38,29)(26,30,36)(28,33,31)$
6A1 $6^{6},1^{2}$ $361$ $6$ $30$ $( 1, 5,18, 8, 4,10)( 2,13, 6, 7,15, 3)( 9,12,17,19,16,11)(20,34,32,35,21,23)(22,31,27,33,24,28)(25,36,29,30,38,26)$
6A-1 $6^{6},1^{2}$ $361$ $6$ $30$ $( 1,10, 4, 8,18, 5)( 2, 3,15, 7, 6,13)( 9,11,16,19,17,12)(20,23,21,35,32,34)(22,28,24,33,27,31)(25,26,38,30,29,36)$
6B1 $6^{6},2$ $361$ $6$ $31$ $( 1,36,11,38, 5,33)( 2,21,18,28,16,20)( 3,25, 6,37, 8,26)( 4,29,13,27,19,32)( 7,22,15,35,14,31)( 9,30,10,34,17,24)(12,23)$
6B-1 $6^{6},2$ $361$ $6$ $31$ $( 1,33, 5,38,11,36)( 2,20,16,28,18,21)( 3,26, 8,37, 6,25)( 4,32,19,27,13,29)( 7,31,14,35,15,22)( 9,24,17,34,10,30)(12,23)$
6C1 $6^{6},2$ $361$ $6$ $31$ $( 1,27,11,25, 5,30)( 2,23,18,35,16,24)( 3,38, 6,26, 8,37)( 4,34,13,36,19,31)( 7,22,15,28,14,32)( 9,33,10,29,17,20)(12,21)$
6C-1 $6^{6},2$ $361$ $6$ $31$ $( 1,30, 5,25,11,27)( 2,24,16,35,18,23)( 3,37, 8,26, 6,38)( 4,31,19,36,13,34)( 7,32,14,28,15,22)( 9,20,17,29,10,33)(12,21)$
9A1 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1,19,15,18,11, 2, 4,12, 6)( 3, 8, 9,13,10,17, 7, 5,16)(20,26,31,32,36,33,21,30,28)(22,34,25,27,35,29,24,23,38)$
9A-1 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1, 6,12, 4, 2,11,18,15,19)( 3,16, 5, 7,17,10,13, 9, 8)(20,28,30,21,33,36,32,31,26)(22,38,23,24,29,35,27,25,34)$
9A2 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1,15,11, 4, 6,19,18, 2,12)( 3, 9,10, 7,16, 8,13,17, 5)(20,31,36,21,28,26,32,33,30)(22,25,35,24,38,34,27,29,23)$
9A-2 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1,12, 2,18,19, 6, 4,11,15)( 3, 5,17,13, 8,16, 7,10, 9)(20,30,33,32,26,28,21,36,31)(22,23,29,27,34,38,24,35,25)$
9A4 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1,11, 6,18,12,15, 4,19, 2)( 3,10,16,13, 5, 9, 7, 8,17)(20,36,28,32,30,31,21,26,33)(22,35,38,27,23,25,24,34,29)$
9A-4 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1, 2,19, 4,15,12,18, 6,11)( 3,17, 8, 7, 9, 5,13,16,10)(20,33,26,21,31,30,32,28,36)(22,29,34,24,25,23,27,38,35)$
18A1 $18^{2},1^{2}$ $361$ $18$ $34$ $( 1, 7,19, 5,15,16,18, 3,11, 8, 2, 9, 4,13,12,10, 6,17)(20,22,26,34,31,25,32,27,36,35,33,29,21,24,30,23,28,38)$
18A-1 $18^{2},1^{2}$ $361$ $18$ $34$ $( 1,17, 6,10,12,13, 4, 9, 2, 8,11, 3,18,16,15, 5,19, 7)(20,38,28,23,30,24,21,29,33,35,36,27,32,25,31,34,26,22)$
18A5 $18^{2},1^{2}$ $361$ $18$ $34$ $( 1,16, 2,10,19, 3, 4,17,15, 8,12, 7,18, 9, 6, 5,11,13)(20,25,33,23,26,27,21,38,31,35,30,22,32,29,28,34,36,24)$
18A-5 $18^{2},1^{2}$ $361$ $18$ $34$ $( 1,13,11, 5, 6, 9,18, 7,12, 8,15,17, 4, 3,19,10, 2,16)(20,24,36,34,28,29,32,22,30,35,31,38,21,27,26,23,33,25)$
18A7 $18^{2},1^{2}$ $361$ $18$ $34$ $( 1, 3,12, 5, 2,17,18,13,19, 8, 6,16, 4, 7,11,10,15, 9)(20,27,30,34,33,38,32,24,26,35,28,25,21,22,36,23,31,29)$
18A-7 $18^{2},1^{2}$ $361$ $18$ $34$ $( 1, 9,15,10,11, 7, 4,16, 6, 8,19,13,18,17, 2, 5,12, 3)(20,29,31,23,36,22,21,25,28,35,26,24,32,38,33,34,30,27)$
18B1 $18^{2},2$ $361$ $18$ $35$ $( 1,35, 8,36,14,26,11,31, 3,38, 7,25, 5,22, 6,33,15,37)( 2,27,17,21,19,24,18,32, 9,28, 4,30,16,29,10,20,13,34)(12,23)$
18B-1 $18^{2},2$ $361$ $18$ $35$ $( 1,37,15,33, 6,22, 5,25, 7,38, 3,31,11,26,14,36, 8,35)( 2,34,13,20,10,29,16,30, 4,28, 9,32,18,24,19,21,17,27)(12,23)$
18B5 $18^{2},2$ $361$ $18$ $35$ $( 1,26, 7,33, 8,31, 5,37,14,38, 6,35,11,25,15,36, 3,22)( 2,24, 4,20,17,32,16,34,19,28,10,27,18,30,13,21, 9,29)(12,23)$
18B-5 $18^{2},2$ $361$ $18$ $35$ $( 1,22, 3,36,15,25,11,35, 6,38,14,37, 5,31, 8,33, 7,26)( 2,29, 9,21,13,30,18,27,10,28,19,34,16,32,17,20, 4,24)(12,23)$
18B7 $18^{2},2$ $361$ $18$ $35$ $( 1,31, 6,36, 7,37,11,22, 8,38,15,26, 5,35, 3,33,14,25)( 2,32,10,21, 4,34,18,29,17,28,13,24,16,27, 9,20,19,30)(12,23)$
18B-7 $18^{2},2$ $361$ $18$ $35$ $( 1,25,14,33, 3,35, 5,26,15,38, 8,22,11,37, 7,36, 6,31)( 2,30,19,20, 9,27,16,24,13,28,17,29,18,34, 4,21,10,32)(12,23)$
18C1 $18^{2},2$ $361$ $18$ $35$ $( 1,28, 8,27,14,37,11,32, 3,25, 7,38, 5,22, 6,30,15,26)( 2,36,17,23,19,20,18,31, 9,35, 4,33,16,34,10,24,13,29)(12,21)$
18C-1 $18^{2},2$ $361$ $18$ $35$ $( 1,26,15,30, 6,22, 5,38, 7,25, 3,32,11,37,14,27, 8,28)( 2,29,13,24,10,34,16,33, 4,35, 9,31,18,20,19,23,17,36)(12,21)$
18C5 $18^{2},2$ $361$ $18$ $35$ $( 1,37, 7,30, 8,32, 5,26,14,25, 6,28,11,38,15,27, 3,22)( 2,20, 4,24,17,31,16,29,19,35,10,36,18,33,13,23, 9,34)(12,21)$
18C-5 $18^{2},2$ $361$ $18$ $35$ $( 1,22, 3,27,15,38,11,28, 6,25,14,26, 5,32, 8,30, 7,37)( 2,34, 9,23,13,33,18,36,10,35,19,29,16,31,17,24, 4,20)(12,21)$
18C7 $18^{2},2$ $361$ $18$ $35$ $( 1,32, 6,27, 7,26,11,22, 8,25,15,37, 5,28, 3,30,14,38)( 2,31,10,23, 4,29,18,34,17,35,13,20,16,36, 9,24,19,33)(12,21)$
18C-7 $18^{2},2$ $361$ $18$ $35$ $( 1,38,14,30, 3,28, 5,37,15,25, 8,22,11,26, 7,27, 6,32)( 2,33,19,24, 9,36,16,20,13,35,17,34,18,29, 4,23,10,31)(12,21)$
19A $19^{2}$ $18$ $19$ $36$ $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,37,35,33,31,29,27,25,23,21,38,36,34,32,30,28,26,24,22)$
19B $19^{2}$ $18$ $19$ $36$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,23,26,29,32,35,38,22,25,28,31,34,37,21,24,27,30,33,36)$
19C $19^{2}$ $36$ $19$ $36$ $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)(20,35,31,27,23,38,34,30,26,22,37,33,29,25,21,36,32,28,24)$
19D $19^{2}$ $36$ $19$ $36$ $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,36,33,30,27,24,21,37,34,31,28,25,22,38,35,32,29,26,23)$
19E $19^{2}$ $36$ $19$ $36$ $( 1,17,14,11, 8, 5, 2,18,15,12, 9, 6, 3,19,16,13,10, 7, 4)(20,30,21,31,22,32,23,33,24,34,25,35,26,36,27,37,28,38,29)$
19F $19^{2}$ $36$ $19$ $36$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,37,35,33,31,29,27,25,23,21,38,36,34,32,30,28,26,24,22)$
19G $19^{2}$ $36$ $19$ $36$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,25,30,35,21,26,31,36,22,27,32,37,23,28,33,38,24,29,34)$
19H $19^{2}$ $36$ $19$ $36$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21)$
19I $19^{2}$ $36$ $19$ $36$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,32,25,37,30,23,35,28,21,33,26,38,31,24,36,29,22,34,27)$
19J $19^{2}$ $36$ $19$ $36$ $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,26,32,38,25,31,37,24,30,36,23,29,35,22,28,34,21,27,33)$
19K $19,1^{19}$ $36$ $19$ $18$ $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)$
38A $38$ $342$ $38$ $37$ $( 1,37, 5,35, 9,33,13,31,17,29, 2,27, 6,25,10,23,14,21,18,38, 3,36, 7,34,11,32,15,30,19,28, 4,26, 8,24,12,22,16,20)$
38B $38$ $342$ $38$ $37$ $( 1,23, 7,26,13,29,19,32, 6,35,12,38,18,22, 5,25,11,28,17,31, 4,34,10,37,16,21, 3,24, 9,27,15,30, 2,33, 8,36,14,20)$

Malle's constant $a(G)$:     $1/18$

Copy content magma:ConjugacyClasses(G);
 

Character table

49 x 49 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed