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Properties

Label 38T22
38T22 1 9 1->9 26 1->26 2 18 2->18 29 2->29 3 8 3->8 32 3->32 4 17 4->17 35 4->35 5 7 5->7 38 5->38 6 16 6->16 22 6->22 7->6 25 7->25 15 8->15 28 8->28 9->5 31 9->31 10 14 10->14 34 10->34 11 11->4 37 11->37 12 13 12->13 21 12->21 13->3 24 13->24 14->12 27 14->27 15->2 30 15->30 16->11 33 16->33 17->1 36 17->36 18->10 20 18->20 19 23 19->23 20->12 20->36 21->5 21->26 22->17 22->35 23->10 23->25 24->3 24->34 25->15 25->24 26->8 26->33 27->1 27->23 28->13 28->32 29->6 29->22 30->18 30->31 31->11 31->21 32->4 32->30 33->20 34->9 34->29 35->2 35->38 36->14 36->28 37->7 38->19 38->27
Degree $38$
Order $6498$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{19}:F_{19}$

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Copy content magma:G := TransitiveGroup(38, 22);
 

Group invariants

Abstract group:  $C_{19}:F_{19}$
Copy content magma:IdentifyGroup(G);
 
Order:  $6498=2 \cdot 3^{2} \cdot 19^{2}$
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 $t$:  $22$
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:  $(1,26,8,28,13,24,3,32,4,35,2,29,6,22,17,36,14,27)(5,38,19,23,10,34,9,31,11,37,7,25,15,30,18,20,12,21)(16,33)$, $(1,9,5,7,6,16,11,4,17)(2,18,10,14,12,13,3,8,15)(20,36,28,32,30,31,21,26,33)(22,35,38,27,23,25,24,34,29)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$9$:  $C_9$
$18$:  $C_{18}$
$171$:  $C_{19}:C_{9}$
$342$:  $F_{19}$, 38T7

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: None

Low degree siblings

38T22 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 $2^{19}$ $19$ $2$ $19$ $( 1,20)( 2,30)( 3,21)( 4,31)( 5,22)( 6,32)( 7,23)( 8,33)( 9,24)(10,34)(11,25)(12,35)(13,26)(14,36)(15,27)(16,37)(17,28)(18,38)(19,29)$
3A1 $3^{12},1^{2}$ $361$ $3$ $24$ $( 1, 6, 4)( 2,17,11)( 3, 9,18)( 5,12,13)( 7,15, 8)(14,16,19)(20,32,31)(21,24,38)(22,35,26)(23,27,33)(25,30,28)(29,36,37)$
3A-1 $3^{12},1^{2}$ $361$ $3$ $24$ $( 1, 4, 6)( 2,11,17)( 3,18, 9)( 5,13,12)( 7, 8,15)(14,19,16)(20,31,32)(21,38,24)(22,26,35)(23,33,27)(25,28,30)(29,37,36)$
6A1 $6^{6},2$ $361$ $6$ $31$ $( 1,31, 6,20, 4,32)( 2,25,17,30,11,28)( 3,38, 9,21,18,24)( 5,26,12,22,13,35)( 7,33,15,23, 8,27)(10,34)(14,29,16,36,19,37)$
6A-1 $6^{6},2$ $361$ $6$ $31$ $( 1,32, 4,20, 6,31)( 2,28,11,30,17,25)( 3,24,18,21, 9,38)( 5,35,13,22,12,26)( 7,27, 8,23,15,33)(10,34)(14,37,19,36,16,29)$
9A1 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1, 9,12, 6,18,13, 4, 3, 5)( 2, 7,16,17,15,19,11, 8,14)(20,24,35,32,38,26,31,21,22)(23,37,28,27,29,25,33,36,30)$
9A-1 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1, 5, 3, 4,13,18, 6,12, 9)( 2,14, 8,11,19,15,17,16, 7)(20,22,21,31,26,38,32,35,24)(23,30,36,33,25,29,27,28,37)$
9A2 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1,12,18, 4, 5, 9, 6,13, 3)( 2,16,15,11,14, 7,17,19, 8)(20,35,38,31,22,24,32,26,21)(23,28,29,33,30,37,27,25,36)$
9A-2 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1, 3,13, 6, 9, 5, 4,18,12)( 2, 8,19,17, 7,14,11,15,16)(20,21,26,32,24,22,31,38,35)(23,36,25,27,37,30,33,29,28)$
9A4 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1,18, 5, 6, 3,12, 4, 9,13)( 2,15,14,17, 8,16,11, 7,19)(20,38,22,32,21,35,31,24,26)(23,29,30,27,36,28,33,37,25)$
9A-4 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1,13, 9, 4,12, 3, 6, 5,18)( 2,19, 7,11,16, 8,17,14,15)(20,26,24,31,35,21,32,22,38)(23,25,37,33,28,36,27,30,29)$
18A1 $18^{2},2$ $361$ $18$ $35$ $( 1,26, 9,31,12,21, 6,22,18,20,13,24, 4,35, 3,32, 5,38)( 2,29, 7,25,16,33,17,36,15,30,19,23,11,37, 8,28,14,27)(10,34)$
18A-1 $18^{2},2$ $361$ $18$ $35$ $( 1,38, 5,32, 3,35, 4,24,13,20,18,22, 6,21,12,31, 9,26)( 2,27,14,28, 8,37,11,23,19,30,15,36,17,33,16,25, 7,29)(10,34)$
18A5 $18^{2},2$ $361$ $18$ $35$ $( 1,21,13,32, 9,22, 4,38,12,20, 3,26, 6,24, 5,31,18,35)( 2,33,19,28, 7,36,11,27,16,30, 8,29,17,23,14,25,15,37)(10,34)$
18A-5 $18^{2},2$ $361$ $18$ $35$ $( 1,35,18,31, 5,24, 6,26, 3,20,12,38, 4,22, 9,32,13,21)( 2,37,15,25,14,23,17,29, 8,30,16,27,11,36, 7,28,19,33)(10,34)$
18A7 $18^{2},2$ $361$ $18$ $35$ $( 1,22, 3,31,13,38, 6,35, 9,20, 5,21, 4,26,18,32,12,24)( 2,36, 8,25,19,27,17,37, 7,30,14,33,11,29,15,28,16,23)(10,34)$
18A-7 $18^{2},2$ $361$ $18$ $35$ $( 1,24,12,32,18,26, 4,21, 5,20, 9,35, 6,38,13,31, 3,22)( 2,23,16,28,15,29,11,33,14,30, 7,37,17,27,19,25, 8,36)(10,34)$
19A1 $19^{2}$ $9$ $19$ $36$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,23,26,29,32,35,38,22,25,28,31,34,37,21,24,27,30,33,36)$
19A-1 $19^{2}$ $9$ $19$ $36$ $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,36,33,30,27,24,21,37,34,31,28,25,22,38,35,32,29,26,23)$
19B $19^{2}$ $18$ $19$ $36$ $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,25,30,35,21,26,31,36,22,27,32,37,23,28,33,38,24,29,34)$
19C1 $19^{2}$ $18$ $19$ $36$ $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,27,34,22,29,36,24,31,38,26,33,21,28,35,23,30,37,25,32)$
19C-1 $19^{2}$ $18$ $19$ $36$ $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$
19D1 $19,1^{19}$ $18$ $19$ $18$ $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
19D-1 $19,1^{19}$ $18$ $19$ $18$ $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)$
19E1 $19^{2}$ $18$ $19$ $36$ $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,32,25,37,30,23,35,28,21,33,26,38,31,24,36,29,22,34,27)$
19E-1 $19^{2}$ $18$ $19$ $36$ $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,37,35,33,31,29,27,25,23,21,38,36,34,32,30,28,26,24,22)$
19F1 $19^{2}$ $18$ $19$ $36$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,27,34,22,29,36,24,31,38,26,33,21,28,35,23,30,37,25,32)$
19F-1 $19^{2}$ $18$ $19$ $36$ $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$
19G1 $19^{2}$ $18$ $19$ $36$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$
19G-1 $19^{2}$ $18$ $19$ $36$ $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,37,35,33,31,29,27,25,23,21,38,36,34,32,30,28,26,24,22)$
19H1 $19^{2}$ $18$ $19$ $36$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,32,25,37,30,23,35,28,21,33,26,38,31,24,36,29,22,34,27)$
19H-1 $19^{2}$ $18$ $19$ $36$ $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,32,25,37,30,23,35,28,21,33,26,38,31,24,36,29,22,34,27)$
19I1 $19^{2}$ $18$ $19$ $36$ $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,27,34,22,29,36,24,31,38,26,33,21,28,35,23,30,37,25,32)$
19I-1 $19^{2}$ $18$ $19$ $36$ $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,35,31,27,23,38,34,30,26,22,37,33,29,25,21,36,32,28,24)$
19J1 $19^{2}$ $18$ $19$ $36$ $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,22,24,26,28,30,32,34,36,38,21,23,25,27,29,31,33,35,37)$
19J-1 $19^{2}$ $18$ $19$ $36$ $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)(20,25,30,35,21,26,31,36,22,27,32,37,23,28,33,38,24,29,34)$
19K1 $19^{2}$ $18$ $19$ $36$ $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)(20,36,33,30,27,24,21,37,34,31,28,25,22,38,35,32,29,26,23)$
19K-1 $19^{2}$ $18$ $19$ $36$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)$
38A1 $38$ $171$ $38$ $37$ $( 1,36, 7,20,13,23,19,26, 6,29,12,32,18,35, 5,38,11,22,17,25, 4,28,10,31,16,34, 3,37, 9,21,15,24, 2,27, 8,30,14,33)$
38A-1 $38$ $171$ $38$ $37$ $( 1,33,14,30, 8,27, 2,24,15,21, 9,37, 3,34,16,31,10,28, 4,25,17,22,11,38, 5,35,18,32,12,29, 6,26,19,23,13,20, 7,36)$

Malle's constant $a(G)$:     $1/18$

Copy content magma:ConjugacyClasses(G);
 

Character table

41 x 41 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed