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Properties

Label 38T38
38T38 1 8 1->8 35 1->35 2 13 2->13 24 2->24 3 18 3->18 32 3->32 4 21 4->21 5 9 5->9 29 5->29 6 14 6->14 37 6->37 7 19 7->19 26 7->26 8->5 34 8->34 10 9->10 23 9->23 15 10->15 31 10->31 11 11->1 20 11->20 12 12->6 28 12->28 13->11 36 13->36 16 14->16 25 14->25 15->2 33 15->33 16->7 22 16->22 17 17->12 30 17->30 18->17 38 18->38 19->3 27 19->27 20->18 21->19 21->29 22->1 22->38 23->2 23->28 24->3 24->37 25->4 25->27 26->5 26->36 27->6 27->26 28->7 28->35 29->8 29->25 30->9 30->34 31->24 32->11 32->33 33->12 33->23 34->13 34->32 35->14 35->22 36->15 36->31 37->16 37->21 38->17 38->30
Degree $38$
Order $38988$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{19}^2:C_9:C_{12}$

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

Group invariants

Abstract group:  $C_{19}^2:C_9:C_{12}$
Copy content magma:IdentifyGroup(G);
 
Order:  $38988=2^{2} \cdot 3^{3} \cdot 19^{2}$
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 $t$:  $38$
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:  $(1,35,14,25,4,21,19,27,6,37,16,22)(2,24,3,32,11,20,18,38,17,30,9,23)(5,29,8,34,13,36,15,33,12,28,7,26)(10,31)$, $(1,8,5,9,10,15,2,13,11)(3,18,17,12,6,14,16,7,19)(21,29,25,27,26,36,31,24,37)(22,38,30,34,32,33,23,28,35)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $S_3$, $C_6$
$12$:  $C_{12}$, $C_3 : C_4$
$18$:  $S_3\times C_3$, $D_{9}$
$36$:  $C_3\times (C_3 : C_4)$, 36T9
$54$:  18T19
$108$:  36T68

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 $2^{18},1^{2}$ $361$ $2$ $18$ $( 1,11)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,19)(13,18)(14,17)(15,16)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(37,38)$
3A1 $3^{6},1^{20}$ $38$ $3$ $12$ $(21,31,27)(22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$
3A-1 $3^{6},1^{20}$ $38$ $3$ $12$ $(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$
3B1 $3^{12},1^{2}$ $361$ $3$ $24$ $( 1,13,12)( 2, 5,19)( 3,16, 7)( 4, 8,14)( 6,11, 9)(10,17,18)(20,36,22)(21,28,29)(23,31,24)(25,34,38)(27,37,33)(30,32,35)$
3B-1 $3^{12},1^{2}$ $361$ $3$ $24$ $( 1,12,13)( 2,19, 5)( 3, 7,16)( 4,14, 8)( 6, 9,11)(10,18,17)(20,22,36)(21,29,28)(23,24,31)(25,38,34)(27,33,37)(30,35,32)$
3C $3^{12},1^{2}$ $722$ $3$ $24$ $( 1, 9, 8)( 2,16,19)( 3, 4,11)( 5,18,14)( 7,13,17)(10,15,12)(20,35,29)(21,27,36)(22,38,24)(23,30,31)(25,33,26)(32,34,37)$
4A1 $4^{9},2$ $3249$ $4$ $28$ $( 1,29,10,23)( 2,22, 9,30)( 3,34, 8,37)( 4,27, 7,25)( 5,20, 6,32)(11,35,19,36)(12,28,18,24)(13,21,17,31)(14,33,16,38)(15,26)$
4A-1 $4^{9},2$ $3249$ $4$ $28$ $( 1,23,10,29)( 2,30, 9,22)( 3,37, 8,34)( 4,25, 7,27)( 5,32, 6,20)(11,36,19,35)(12,24,18,28)(13,31,17,21)(14,38,16,33)(15,26)$
6A1 $6^{6},1^{2}$ $361$ $6$ $30$ $( 1,18,13,10,12,17)( 2,11, 5, 9,19, 6)( 3, 4,16, 8, 7,14)(20,30,36,32,22,35)(21,23,28,31,29,24)(25,33,34,27,38,37)$
6A-1 $6^{6},1^{2}$ $361$ $6$ $30$ $( 1,17,12,10,13,18)( 2, 6,19, 9, 5,11)( 3,14, 7, 8,16, 4)(20,35,22,32,36,30)(21,24,29,31,28,23)(25,37,38,27,34,33)$
6B $6^{6},1^{2}$ $722$ $6$ $30$ $( 1, 4, 9,11, 8, 3)( 2,12,16,10,19,15)( 5,17,18, 7,14,13)(20,27,35,36,29,21)(22,32,38,34,24,37)(23,25,30,33,31,26)$
6C1 $6^{3},2^{9},1^{2}$ $722$ $6$ $24$ $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(21,32,31,38,27,28)(22,25,23,37,34,36)(24,30,26,35,29,33)$
6C-1 $6^{3},2^{9},1^{2}$ $722$ $6$ $24$ $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(21,28,27,38,31,32)(22,36,34,37,23,25)(24,33,29,35,26,30)$
9A1 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1,17, 2, 3, 8,14, 6, 4,13)( 5,18, 7, 9,19,12,15,11,10)(20,21,25,22,29,38,36,28,34)(23,33,35,24,37,32,31,27,30)$
9A2 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1, 2, 8, 6,13,17, 3,14, 4)( 5, 7,19,15,10,18, 9,12,11)(20,25,29,36,34,21,22,38,28)(23,35,37,31,30,33,24,32,27)$
9A4 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1, 8,13, 3, 4, 2, 6,17,14)( 5,19,10, 9,11, 7,15,18,12)(20,29,34,22,28,25,36,21,38)(23,37,30,24,27,35,31,33,32)$
9B1 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1,18,19, 9,14, 2, 8, 5,16)( 3,17,10, 4, 7,15,11,13,12)(20,33,32,35,26,34,29,25,37)(21,30,22,27,31,38,36,23,24)$
9B-1 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1,16, 5, 8, 2,14, 9,19,18)( 3,12,13,11,15, 7, 4,10,17)(20,37,25,29,34,26,35,32,33)(21,24,23,36,38,31,27,22,30)$
9B2 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1,19,14, 8,16,18, 9, 2, 5)( 3,10, 7,11,12,17, 4,15,13)(20,32,26,29,37,33,35,34,25)(21,22,31,36,24,30,27,38,23)$
9B-2 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1, 5, 2, 9,18,16, 8,14,19)( 3,13,15, 4,17,12,11, 7,10)(20,25,34,35,33,37,29,26,32)(21,23,38,27,30,24,36,31,22)$
9B4 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1,14,16, 9, 5,19, 8,18, 2)( 3, 7,12, 4,13,10,11,17,15)(20,26,37,35,25,32,29,33,34)(21,31,24,27,23,22,36,30,38)$
9B-4 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1, 2,18, 8,19, 5, 9,16,14)( 3,15,17,11,10,13, 4,12, 7)(20,34,33,29,32,25,35,37,26)(21,38,30,36,22,23,27,24,31)$
12A1 $12^{3},2$ $3249$ $12$ $34$ $( 1,31,18,29,13,24,10,21,12,23,17,28)( 2,32,11,22, 5,35, 9,20,19,30, 6,36)( 3,33, 4,34,16,27, 8,38, 7,37,14,25)(15,26)$
12A-1 $12^{3},2$ $3249$ $12$ $34$ $( 1,28,17,23,12,21,10,24,13,29,18,31)( 2,36, 6,30,19,20, 9,35, 5,22,11,32)( 3,25,14,37, 7,38, 8,27,16,34, 4,33)(15,26)$
12A5 $12^{3},2$ $3249$ $12$ $34$ $( 1,24,17,29,12,31,10,28,13,23,18,21)( 2,35, 6,22,19,32, 9,36, 5,30,11,20)( 3,27,14,34, 7,33, 8,25,16,37, 4,38)(15,26)$
12A-5 $12^{3},2$ $3249$ $12$ $34$ $( 1,21,18,23,13,28,10,31,12,29,17,24)( 2,20,11,30, 5,36, 9,32,19,22, 6,35)( 3,38, 4,37,16,25, 8,33, 7,34,14,27)(15,26)$
18A1 $18^{2},1^{2}$ $722$ $18$ $34$ $( 1,18,17, 7, 2, 9, 3,19, 8,12,14,15, 6,11, 4,10,13, 5)(20,33,21,35,25,24,22,37,29,32,38,31,36,27,28,30,34,23)$
18A5 $18^{2},1^{2}$ $722$ $18$ $34$ $( 1, 9,14,10,17,19, 6, 5, 2,12, 4,18, 3,15,13, 7, 8,11)(20,24,38,30,21,37,36,23,25,32,28,33,22,31,34,35,29,27)$
18A7 $18^{2},1^{2}$ $722$ $18$ $34$ $( 1,19, 4, 7,14, 5, 3,11,17,12,13, 9, 6,18, 8,10, 2,15)(20,37,28,35,38,23,22,27,21,32,34,24,36,33,29,30,25,31)$
18B1 $18^{2},1^{2}$ $722$ $18$ $34$ $( 1,10,18, 4,19, 7, 9,15,14,11, 2,13, 8,12, 5, 3,16,17)(20,22,33,27,32,31,35,38,26,36,34,23,29,24,25,21,37,30)$
18B-1 $18^{2},1^{2}$ $722$ $18$ $34$ $( 1,17,16, 3, 5,12, 8,13, 2,11,14,15, 9, 7,19, 4,18,10)(20,30,37,21,25,24,29,23,34,36,26,38,35,31,32,27,33,22)$
18B5 $18^{2},1^{2}$ $722$ $18$ $34$ $( 1, 7, 2, 3,18,15, 8,17,19,11, 5,10, 9,13,16, 4,14,12)(20,31,34,21,33,38,29,30,32,36,25,22,35,23,37,27,26,24)$
18B-5 $18^{2},1^{2}$ $722$ $18$ $34$ $( 1,12,14, 4,16,13, 9,10, 5,11,19,17, 8,15,18, 3, 2, 7)(20,24,26,27,37,23,35,22,25,36,32,30,29,38,33,21,34,31)$
18B7 $18^{2},1^{2}$ $722$ $18$ $34$ $( 1,15, 5, 4, 2,17, 9,12,18,11,16, 7, 8,10,14, 3,19,13)(20,38,25,27,34,30,35,24,33,36,37,31,29,22,26,21,32,23)$
18B-7 $18^{2},1^{2}$ $722$ $18$ $34$ $( 1,13,19, 3,14,10, 8, 7,16,11,18,12, 9,17, 2, 4, 5,15)(20,23,32,21,26,22,29,31,37,36,33,24,35,30,34,27,25,38)$
19A $19,1^{19}$ $36$ $19$ $18$ $(20,27,34,22,29,36,24,31,38,26,33,21,28,35,23,30,37,25,32)$
19B1 $19^{2}$ $108$ $19$ $36$ $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$
19B2 $19^{2}$ $108$ $19$ $36$ $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,28,36,25,33,22,30,38,27,35,24,32,21,29,37,26,34,23,31)$
19B4 $19^{2}$ $108$ $19$ $36$ $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,28,36,25,33,22,30,38,27,35,24,32,21,29,37,26,34,23,31)$
57A1 $19,3^{6},1$ $684$ $57$ $30$ $( 1,11, 5)( 2,18,16)( 3, 6, 8)( 4,13,19)( 7,15,14)( 9,10,17)(20,35,31,27,23,38,34,30,26,22,37,33,29,25,21,36,32,28,24)$
57A-1 $19,3^{6},1$ $684$ $57$ $30$ $( 1, 5,11)( 2,16,18)( 3, 8, 6)( 4,19,13)( 7,14,15)( 9,17,10)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

42 x 42 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed