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Properties

Label 38T34
38T34 1 4 1->4 34 1->34 2 10 2->10 28 2->28 3 16 3->16 22 3->22 4->3 35 4->35 5 9 5->9 29 5->29 6 15 6->15 23 6->23 7 7->2 36 7->36 8 30 8->30 14 9->14 24 9->24 10->1 37 10->37 11 11->7 31 11->31 12 13 12->13 25 12->25 19 13->19 38 13->38 14->6 32 14->32 15->12 26 15->26 18 16->18 20 16->20 17 17->5 33 17->33 18->11 27 18->27 19->17 21 19->21 20->13 20->22 21->16 21->25 22->19 22->28 23->3 23->31 24->6 24->34 25->9 25->37 26->12 26->21 27->15 27->24 28->18 28->27 29->2 29->30 30->5 30->33 31->8 31->36 32->11 32->20 33->14 33->23 34->17 34->26 35->1 35->29 36->4 36->32 37->7 37->35 38->10
Degree $38$
Order $25992$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{19}^2:D_9$

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Copy content magma:G := TransitiveGroup(38, 34);
 

Group invariants

Abstract group:  $D_{19}^2:D_9$
Copy content magma:IdentifyGroup(G);
 
Order:  $25992=2^{3} \cdot 3^{2} \cdot 19^{2}$
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 $t$:  $34$
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:  $(1,4,3,16,18,11,7,2,10)(5,9,14,6,15,12,13,19,17)(20,22,28,27,24,34,26,21,25,37,35,29,30,33,23,31,36,32)$, $(1,34,17,33,14,32,11,31,8,30,5,29,2,28,18,27,15,26,12,25,9,24,6,23,3,22,19,21,16,20,13,38,10,37,7,36,4,35)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$
$12$:  $D_{6}$
$18$:  $D_{9}$
$24$:  $(C_6\times C_2):C_2$
$36$:  $D_{18}$
$72$:  36T24

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 $2^{9},1^{20}$ $38$ $2$ $9$ $(20,26)(21,25)(22,24)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)$
2B $2^{19}$ $342$ $2$ $19$ $( 1,31)( 2,22)( 3,32)( 4,23)( 5,33)( 6,24)( 7,34)( 8,25)( 9,35)(10,26)(11,36)(12,27)(13,37)(14,28)(15,38)(16,29)(17,20)(18,30)(19,21)$
2C $2^{18},1^{2}$ $361$ $2$ $18$ $( 1,19)( 2,18)( 3,17)( 4,16)( 5,15)( 6,14)( 7,13)( 8,12)( 9,11)(20,25)(21,24)(22,23)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)$
3A $3^{12},1^{2}$ $722$ $3$ $24$ $( 1,13,12)( 2, 5,19)( 3,16, 7)( 4, 8,14)( 6,11, 9)(10,17,18)(20,21,28)(22,35,31)(24,30,34)(25,37,26)(27,32,29)(33,36,38)$
4A $4^{9},2$ $6498$ $4$ $28$ $( 1,22,19,23)( 2,21,18,24)( 3,20,17,25)( 4,38,16,26)( 5,37,15,27)( 6,36,14,28)( 7,35,13,29)( 8,34,12,30)( 9,33,11,31)(10,32)$
6A $6^{6},1^{2}$ $722$ $6$ $30$ $( 1,18, 2, 7, 9, 6)( 3,15,16, 5,12,11)( 8,17,13,19,10,14)(20,24,34,21,36,26)(22,29,37,38,31,23)(25,27,32,35,33,28)$
6B1 $6^{3},3^{6},1^{2}$ $722$ $6$ $27$ $( 1,12,13)( 2,19, 5)( 3, 7,16)( 4,14, 8)( 6, 9,11)(10,18,17)(20,37,21,26,28,25)(22,34,35,24,31,30)(27,36,32,38,29,33)$
6B-1 $6^{3},3^{6},1^{2}$ $722$ $6$ $27$ $( 1,13,12)( 2, 5,19)( 3,16, 7)( 4, 8,14)( 6,11, 9)(10,17,18)(20,25,28,26,21,37)(22,30,31,24,35,34)(27,33,29,38,32,36)$
9A1 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1, 2, 7,13, 5, 3,12,19,16)( 4,17, 6, 8,18,11,14,10, 9)(20,30,32,21,34,29,28,24,27)(22,38,26,35,33,25,31,36,37)$
9A2 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1, 7, 5,12,16, 2,13, 3,19)( 4, 6,18,14, 9,17, 8,11,10)(20,32,34,28,27,30,21,29,24)(22,26,33,31,37,38,35,25,36)$
9A4 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1, 5,16,13,19, 7,12, 2, 3)( 4,18, 9, 8,10, 6,14,17,11)(20,34,27,21,24,32,28,30,29)(22,33,37,35,36,26,31,38,25)$
18A1 $18^{2},1^{2}$ $722$ $18$ $34$ $( 1,19, 5,18,10,12, 2,14,11, 7, 8, 3, 9,17,15, 6,13,16)(20,32,22,24,35,29,34,33,37,21,28,38,36,25,31,26,27,23)$
18A5 $18^{2},1^{2}$ $722$ $18$ $34$ $( 1,12, 8, 6, 5,14, 9,16,10, 7,15,19, 2, 3,13,18,11,17)(20,29,28,26,22,33,36,23,35,21,31,32,34,38,27,24,37,25)$
18A7 $18^{2},1^{2}$ $722$ $18$ $34$ $( 1,14,15,18, 8,16, 2,17, 5, 7,13,12, 9,19,11, 6,10, 3)(20,33,31,24,28,23,34,25,22,21,27,29,36,32,37,26,35,38)$
18B1 $18,9^{2},1^{2}$ $722$ $18$ $33$ $( 1, 3, 2,12, 7,19,13,16, 5)( 4,11,17,14, 6,10, 8, 9,18)(20,36,30,37,32,22,21,38,34,26,29,35,28,33,24,25,27,31)$
18B-1 $18,9^{2},1^{2}$ $722$ $18$ $33$ $( 1, 5,16,13,19, 7,12, 2, 3)( 4,18, 9, 8,10, 6,14,17,11)(20,31,27,25,24,33,28,35,29,26,34,38,21,22,32,37,30,36)$
18B5 $18,9^{2},1^{2}$ $722$ $18$ $33$ $( 1,19, 3,13, 2,16,12, 5, 7)( 4,10,11, 8,17, 9,14,18, 6)(20,22,29,25,30,38,28,31,32,26,24,36,21,35,27,37,34,33)$
18B-5 $18,9^{2},1^{2}$ $722$ $18$ $33$ $( 1, 7, 5,12,16, 2,13, 3,19)( 4, 6,18,14, 9,17, 8,11,10)(20,33,34,37,27,35,21,36,24,26,32,31,28,38,30,25,29,22)$
18B7 $18,9^{2},1^{2}$ $722$ $18$ $33$ $( 1,16,19,12, 3, 5,13, 7, 2)( 4, 9,10,14,11,18, 8, 6,17)(20,38,24,37,29,31,21,33,30,26,27,22,28,36,34,25,32,35)$
18B-7 $18,9^{2},1^{2}$ $722$ $18$ $33$ $( 1, 2, 7,13, 5, 3,12,19,16)( 4,17, 6, 8,18,11,14,10, 9)(20,35,32,25,34,36,28,22,27,26,30,33,21,31,29,37,24,38)$
19A $19,1^{19}$ $36$ $19$ $18$ $(20,25,30,35,21,26,31,36,22,27,32,37,23,28,33,38,24,29,34)$
19B1 $19^{2}$ $36$ $19$ $36$ $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,36,33,30,27,24,21,37,34,31,28,25,22,38,35,32,29,26,23)$
19B2 $19^{2}$ $36$ $19$ $36$ $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)(20,33,27,21,34,28,22,35,29,23,36,30,24,37,31,25,38,32,26)$
19B3 $19^{2}$ $36$ $19$ $36$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,30,21,31,22,32,23,33,24,34,25,35,26,36,27,37,28,38,29)$
19B4 $19^{2}$ $36$ $19$ $36$ $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,27,34,22,29,36,24,31,38,26,33,21,28,35,23,30,37,25,32)$
19B5 $19^{2}$ $36$ $19$ $36$ $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$
19B6 $19^{2}$ $36$ $19$ $36$ $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)$
19B7 $19^{2}$ $36$ $19$ $36$ $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,37,35,33,31,29,27,25,23,21,38,36,34,32,30,28,26,24,22)$
19B8 $19^{2}$ $36$ $19$ $36$ $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$
19B9 $19^{2}$ $36$ $19$ $36$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,31,23,34,26,37,29,21,32,24,35,27,38,30,22,33,25,36,28)$
38A $19,2^{9},1$ $684$ $38$ $27$ $( 1,13)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(14,19)(15,18)(16,17)(20,32,25,37,30,23,35,28,21,33,26,38,31,24,36,29,22,34,27)$
38B1 $38$ $684$ $38$ $37$ $( 1,20,14,36, 8,33, 2,30,15,27, 9,24, 3,21,16,37,10,34, 4,31,17,28,11,25, 5,22,18,38,12,35, 6,32,19,29,13,26, 7,23)$
38B3 $38$ $684$ $38$ $37$ $( 1,36, 2,27, 3,37, 4,28, 5,38, 6,29, 7,20, 8,30, 9,21,10,31,11,22,12,32,13,23,14,33,15,24,16,34,17,25,18,35,19,26)$
38B5 $38$ $684$ $38$ $37$ $( 1,33, 9,37,17,22, 6,26,14,30, 3,34,11,38,19,23, 8,27,16,31, 5,35,13,20, 2,24,10,28,18,32, 7,36,15,21, 4,25,12,29)$
38B7 $38$ $684$ $38$ $37$ $( 1,30,16,28,12,26, 8,24, 4,22,19,20,15,37,11,35, 7,33, 3,31,18,29,14,27,10,25, 6,23, 2,21,17,38,13,36, 9,34, 5,32)$
38B9 $38$ $684$ $38$ $37$ $( 1,27, 4,38, 7,30,10,22,13,33,16,25,19,36, 3,28, 6,20, 9,31,12,23,15,34,18,26, 2,37, 5,29, 8,21,11,32,14,24,17,35)$
38B11 $38$ $684$ $38$ $37$ $( 1,24,11,29, 2,34,12,20, 3,25,13,30, 4,35,14,21, 5,26,15,31, 6,36,16,22, 7,27,17,32, 8,37,18,23, 9,28,19,33,10,38)$
38B13 $38$ $684$ $38$ $37$ $( 1,21,18,20,16,38,14,37,12,36,10,35, 8,34, 6,33, 4,32, 2,31,19,30,17,29,15,28,13,27,11,26, 9,25, 7,24, 5,23, 3,22)$
38B15 $38$ $684$ $38$ $37$ $( 1,37, 6,30,11,23,16,35, 2,28, 7,21,12,33,17,26, 3,38, 8,31,13,24,18,36, 4,29, 9,22,14,34,19,27, 5,20,10,32,15,25)$
38B17 $38$ $684$ $38$ $37$ $( 1,34,13,21, 6,27,18,33,11,20, 4,26,16,32, 9,38, 2,25,14,31, 7,37,19,24,12,30, 5,36,17,23,10,29, 3,35,15,22, 8,28)$

Malle's constant $a(G)$:     $1/9$

Copy content magma:ConjugacyClasses(G);
 

Character table

41 x 41 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed