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Properties

Label 38T32
38T32 1 2 1->2 26 1->26 6 2->6 22 2->22 3 10 3->10 37 3->37 4 14 4->14 33 4->33 5 18 5->18 29 5->29 6->3 25 6->25 7 21 7->21 8 11 8->11 36 8->36 9 15 9->15 32 9->32 19 10->19 28 10->28 11->4 24 11->24 12 12->8 20 12->20 13 13->12 35 13->35 16 14->16 31 14->31 15->1 27 15->27 16->5 23 16->23 17 17->9 38 17->38 18->13 34 18->34 19->17 30 19->30 20->19 20->27 21->3 21->33 22->6 22->20 23->9 23->26 24->12 24->32 25->15 25->38 26->18 26->25 27->2 27->31 28->5 28->37 29->8 29->24 30->11 31->36 32->17 32->23 33->1 33->29 34->4 34->35 35->7 35->22 36->10 36->28 37->13 37->34 38->16 38->21
Degree $38$
Order $19494$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{19}^2:(S_3\times C_9)$

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Copy content magma:G := TransitiveGroup(38, 32);
 

Group invariants

Abstract group:  $C_{19}^2:(S_3\times C_9)$
Copy content magma:IdentifyGroup(G);
 
Order:  $19494=2 \cdot 3^{3} \cdot 19^{2}$
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 $t$:  $32$
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:  $(1,26,18,34,4,33)(2,22,6,25,15,27)(3,37,13,35,7,21)(5,29,8,36,10,28)(9,32,17,38,16,23)(11,24,12,20,19,30)(14,31)$, $(1,2,6,3,10,19,17,9,15)(4,14,16,5,18,13,12,8,11)(20,27,31,36,28,37,34,35,22)(21,33,29,24,32,23,26,25,38)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$9$:  $C_9$
$18$:  $S_3\times C_3$, $C_{18}$
$54$:  $C_9\times S_3$

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 $2^{19}$ $57$ $2$ $19$ $( 1,29)( 2,20)( 3,30)( 4,21)( 5,31)( 6,22)( 7,32)( 8,23)( 9,33)(10,24)(11,34)(12,25)(13,35)(14,26)(15,36)(16,27)(17,37)(18,28)(19,38)$
3A1 $3^{6},1^{20}$ $38$ $3$ $12$ $(20,24,30)(21,35,37)(22,27,25)(23,38,32)(26,33,34)(28,36,29)$
3A-1 $3^{6},1^{20}$ $38$ $3$ $12$ $(20,30,24)(21,37,35)(22,25,27)(23,32,38)(26,34,33)(28,29,36)$
3B1 $3^{12},1^{2}$ $361$ $3$ $24$ $( 1, 7,16)( 2,18, 4)( 3,10,11)( 5,13, 6)( 9,19,15)(12,14,17)(20,31,38)(21,23,26)(22,34,33)(24,37,28)(25,29,35)(27,32,30)$
3B-1 $3^{12},1^{2}$ $361$ $3$ $24$ $( 1,16, 7)( 2, 4,18)( 3,11,10)( 5, 6,13)( 9,15,19)(12,17,14)(20,38,31)(21,26,23)(22,33,34)(24,28,37)(25,35,29)(27,30,32)$
3C $3^{12},1^{2}$ $722$ $3$ $24$ $( 1,14, 5)( 2, 6,12)( 3,17,19)( 4, 9, 7)( 8,15,16)(10,18,11)(20,32,21)(22,27,24)(23,34,35)(25,29,38)(26,36,30)(28,31,33)$
6A1 $6^{6},2$ $1083$ $6$ $31$ $( 1,23, 7,26,16,21)( 2,33,18,22, 4,34)( 3,24,10,37,11,28)( 5,25,13,29, 6,35)( 8,36)( 9,27,19,32,15,30)(12,38,14,20,17,31)$
6A-1 $6^{6},2$ $1083$ $6$ $31$ $( 1,21,16,26, 7,23)( 2,34, 4,22,18,33)( 3,28,11,37,10,24)( 5,35, 6,29,13,25)( 8,36)( 9,30,15,32,19,27)(12,31,17,20,14,38)$
9A1 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1,11, 4, 7, 3, 2,16,10,18)( 5,12, 9,13,14,19, 6,17,15)(20,32,35,31,30,25,38,27,29)(21,37,22,23,28,34,26,24,33)$
9A-1 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1,18,10,16, 2, 3, 7, 4,11)( 5,15,17, 6,19,14,13, 9,12)(20,29,27,38,25,30,31,35,32)(21,33,24,26,34,28,23,22,37)$
9A2 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1, 4, 3,16,18,11, 7, 2,10)( 5, 9,14, 6,15,12,13,19,17)(20,35,30,38,29,32,31,25,27)(21,22,28,26,33,37,23,34,24)$
9A-2 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1,10, 2, 7,11,18,16, 3, 4)( 5,17,19,13,12,15, 6,14, 9)(20,27,25,31,32,29,38,30,35)(21,24,34,23,37,33,26,28,22)$
9A4 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1, 3,18, 7,10, 4,16,11, 2)( 5,14,15,13,17, 9, 6,12,19)(20,30,29,31,27,35,38,32,25)(21,28,33,23,24,22,26,37,34)$
9A-4 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1, 2,11,16, 4,10, 7,18, 3)( 5,19,12, 6, 9,17,13,15,14)(20,25,32,38,35,27,31,29,30)(21,34,37,26,22,24,23,33,28)$
9B1 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1,14, 9, 8, 4, 7,19,10,12)( 2,18, 6,15,13, 5,11,16,17)(20,35,37,36,27,22,34,28,31)(21,25,23,24,33,38,26,32,29)$
9B-1 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1,12,10,19, 7, 4, 8, 9,14)( 2,17,16,11, 5,13,15, 6,18)(20,31,28,34,22,27,36,37,35)(21,29,32,26,38,33,24,23,25)$
9B2 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1, 9, 4,19,12,14, 8, 7,10)( 2, 6,13,11,17,18,15, 5,16)(20,37,27,34,31,35,36,22,28)(21,23,33,26,29,25,24,38,32)$
9B-2 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1,10, 7, 8,14,12,19, 4, 9)( 2,16, 5,15,18,17,11,13, 6)(20,28,22,36,35,31,34,27,37)(21,32,38,24,25,29,26,33,23)$
9B4 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1, 4,12, 8,10, 9,19,14, 7)( 2,13,17,15,16, 6,11,18, 5)(20,27,31,36,28,37,34,35,22)(21,33,29,24,32,23,26,25,38)$
9B-4 $9^{4},1^{2}$ $722$ $9$ $32$ $( 1, 7,14,19, 9,10, 8,12, 4)( 2, 5,18,11, 6,16,15,17,13)(20,22,35,34,37,28,36,31,27)(21,38,25,26,23,32,24,29,33)$
18A1 $18^{2},2$ $1083$ $18$ $35$ $( 1,22,11,23, 4,28, 7,34, 3,26, 2,24,16,33,10,21,18,37)( 5,30,12,25, 9,38,13,27,14,29,19,20, 6,32,17,35,15,31)( 8,36)$
18A-1 $18^{2},2$ $1083$ $18$ $35$ $( 1,37,18,21,10,33,16,24, 2,26, 3,34, 7,28, 4,23,11,22)( 5,31,15,35,17,32, 6,20,19,29,14,27,13,38, 9,25,12,30)( 8,36)$
18A5 $18^{2},2$ $1083$ $18$ $35$ $( 1,28, 2,21,11,34,16,37, 4,26,10,22, 7,24,18,23, 3,33)( 5,38,19,35,12,27, 6,31, 9,29,17,30,13,20,15,25,14,32)( 8,36)$
18A-5 $18^{2},2$ $1083$ $18$ $35$ $( 1,33, 3,23,18,24, 7,22,10,26, 4,37,16,34,11,21, 2,28)( 5,32,14,25,15,20,13,30,17,29, 9,31, 6,27,12,35,19,38)( 8,36)$
18A7 $18^{2},2$ $1083$ $18$ $35$ $( 1,34,10,23, 2,37, 7,33,11,26,18,28,16,22, 3,21, 4,24)( 5,27,17,25,19,31,13,32,12,29,15,38, 6,30,14,35, 9,20)( 8,36)$
18A-7 $18^{2},2$ $1083$ $18$ $35$ $( 1,24, 4,21, 3,22,16,28,18,26,11,33, 7,37, 2,23,10,34)( 5,20, 9,35,14,30, 6,38,15,29,12,32,13,31,19,25,17,27)( 8,36)$
19A1 $19,1^{19}$ $18$ $19$ $18$ $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)$
19A-1 $19,1^{19}$ $18$ $19$ $18$ $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)$
19B1 $19^{2}$ $27$ $19$ $36$ $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)$
19B-1 $19^{2}$ $27$ $19$ $36$ $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21)$
19C $19^{2}$ $54$ $19$ $36$ $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,33,27,21,34,28,22,35,29,23,36,30,24,37,31,25,38,32,26)$
19D1 $19^{2}$ $54$ $19$ $36$ $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,29,38,28,37,27,36,26,35,25,34,24,33,23,32,22,31,21,30)$
19D-1 $19^{2}$ $54$ $19$ $36$ $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21)$
19E1 $19^{2}$ $54$ $19$ $36$ $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21)$
19E-1 $19^{2}$ $54$ $19$ $36$ $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,37,35,33,31,29,27,25,23,21,38,36,34,32,30,28,26,24,22)$
38A1 $38$ $513$ $38$ $37$ $( 1,20, 3,21, 5,22, 7,23, 9,24,11,25,13,26,15,27,17,28,19,29, 2,30, 4,31, 6,32, 8,33,10,34,12,35,14,36,16,37,18,38)$
38A-1 $38$ $513$ $38$ $37$ $( 1,38,18,37,16,36,14,35,12,34,10,33, 8,32, 6,31, 4,30, 2,29,19,28,17,27,15,26,13,25,11,24, 9,23, 7,22, 5,21, 3,20)$
57A1 $19,3^{6},1$ $342$ $57$ $30$ $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,24,30)(21,35,37)(22,27,25)(23,38,32)(26,33,34)(28,36,29)$
57A-1 $19,3^{6},1$ $342$ $57$ $30$ $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,30,24)(21,37,35)(22,25,27)(23,32,38)(26,34,33)(28,29,36)$
57A5 $19,3^{6},1$ $342$ $57$ $30$ $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)(20,30,24)(21,37,35)(22,25,27)(23,32,38)(26,34,33)(28,29,36)$
57A-5 $19,3^{6},1$ $342$ $57$ $30$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,24,30)(21,35,37)(22,27,25)(23,38,32)(26,33,34)(28,36,29)$

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