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Properties

Label 38T41
38T41 1 21 1->21 2 2->1 3 7 3->7 3->21 4 9 4->9 5 11 5->11 13 5->13 6 28 6->28 30 6->30 20 7->20 34 7->34 8 8->9 16 8->16 10 9->10 36 9->36 10->13 26 10->26 11->4 11->5 12 14 12->14 25 12->25 13->11 35 13->35 14->8 17 14->17 15 19 15->19 15->35 29 16->29 16->29 17->12 17->12 18 18->3 18->10 19->6 22 19->22 20->3 38 20->38 21->2 21->20 24 22->24 22->26 23 23->5 31 23->31 24->7 24->19 25->23 33 25->33 26->14 26->18 27 27->15 27->28 28->31 28->38 29->24 29->33 30->36 37 30->37 31->25 32 31->32 32->37 33->16 33->27 34->15 34->17 35->22 35->34 36->8 36->18 37->6 37->30 38->23 38->27
Degree $38$
Order $50616$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\PGL(2,37)$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(38, 41);
 

Group invariants

Abstract group:  $\PGL(2,37)$
Copy content magma:IdentifyGroup(G);
 
Order:  $50616=2^{3} \cdot 3^{2} \cdot 19 \cdot 37$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
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 $t$:  $41$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  yes
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(3,21,20,38,23,5,11,4,9,10,13,35,22,26,14,8,16,29,24,7,34,17,12,25,33,27,15,19,6,28,31,32,37,30,36,18)$, $(1,21,2)(3,7,20)(5,13,11)(6,30,37)(8,9,36)(10,26,18)(12,14,17)(15,35,34)(16,29,33)(19,22,24)(23,31,25)(27,28,38)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

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 $2^{19}$ $666$ $2$ $19$ $( 1,37)( 2,12)( 3,31)( 4, 6)( 5,16)( 7,32)( 8,28)( 9,21)(10,19)(11,36)(13,29)(14,30)(15,35)(17,23)(18,25)(20,26)(22,34)(24,33)(27,38)$
2B $2^{18},1^{2}$ $703$ $2$ $18$ $( 1,20)( 2,27)( 3,35)( 4,18)( 5,37)( 6,34)( 7,11)( 8,28)( 9,17)(10,15)(12,32)(13,38)(16,19)(21,24)(22,36)(23,31)(25,30)(29,33)$
3A $3^{12},1^{2}$ $1406$ $3$ $24$ $( 1, 3,37)( 2,32,19)( 4,11,30)( 5,20,35)( 6,23, 9)( 7,25,18)( 8,38,21)(10,29,36)(12,16,27)(13,24,28)(15,33,22)(17,34,31)$
4A $4^{9},1^{2}$ $1406$ $4$ $27$ $( 1, 6,20,34)( 2, 4,27,18)( 3,23,35,31)( 5,17,37, 9)( 7,32,11,12)( 8,33,28,29)(10,21,15,24)(13,36,38,22)(16,25,19,30)$
6A $6^{6},1^{2}$ $1406$ $6$ $30$ $( 1, 5, 3,20,37,35)( 2,16,32,27,19,12)( 4,25,11,18,30, 7)( 6,17,23,34, 9,31)( 8,24,38,28,21,13)(10,22,29,15,36,33)$
9A1 $9^{4},1^{2}$ $1406$ $9$ $32$ $( 1,29,30, 3,36, 4,37,10,11)( 2,17,24,32,34,28,19,31,13)( 5,15, 7,20,33,25,35,22,18)( 6, 8,16,23,38,27, 9,21,12)$
9A2 $9^{4},1^{2}$ $1406$ $9$ $32$ $( 1,30,36,37,11,29, 3, 4,10)( 2,24,34,19,13,17,32,28,31)( 5, 7,33,35,18,15,20,25,22)( 6,16,38, 9,12, 8,23,27,21)$
9A4 $9^{4},1^{2}$ $1406$ $9$ $32$ $( 1,36,11, 3,10,30,37,29, 4)( 2,34,13,32,31,24,19,17,28)( 5,33,18,20,22, 7,35,15,25)( 6,38,12,23,21,16, 9, 8,27)$
12A1 $12^{3},1^{2}$ $1406$ $12$ $33$ $( 1,31, 5, 6, 3,17,20,23,37,34,35, 9)( 2, 7,16, 4,32,25,27,11,19,18,12,30)( 8,36,24,33,38,10,28,22,21,29,13,15)$
12A5 $12^{3},1^{2}$ $1406$ $12$ $33$ $( 1,17,35, 6,37,31,20, 9, 3,34, 5,23)( 2,25,12, 4,19, 7,27,30,32,18,16,11)( 8,10,13,33,21,36,28,15,38,29,24,22)$
18A1 $18^{2},1^{2}$ $1406$ $18$ $34$ $( 1,18,29, 5,30,15, 3, 7,36,20, 4,33,37,25,10,35,11,22)( 2, 8,17,16,24,23,32,38,34,27,28, 9,19,21,31,12,13, 6)$
18A5 $18^{2},1^{2}$ $1406$ $18$ $34$ $( 1,15, 4,35,29, 7,37,22,30,20,10,18, 3,33,11, 5,36,25)( 2,23,28,12,17,38,19, 6,24,27,31, 8,32, 9,13,16,34,21)$
18A7 $18^{2},1^{2}$ $1406$ $18$ $34$ $( 1, 7,10, 5, 4,22, 3,25,29,20,11,15,37,18,36,35,30,33)( 2,38,31,16,28, 6,32,21,17,27,13,23,19, 8,34,12,24, 9)$
19A1 $19^{2}$ $1332$ $19$ $36$ $( 1,36,16,31,21,17,19,30,29,33,20,34, 2,38,32, 6,15,25, 8)( 3, 9,23,10,14,13,24,26,22,12,27, 7, 4,35,18,28,37,11, 5)$
19A2 $19^{2}$ $1332$ $19$ $36$ $( 1,16,21,19,29,20, 2,32,15, 8,36,31,17,30,33,34,38, 6,25)( 3,23,14,24,22,27, 4,18,37, 5, 9,10,13,26,12, 7,35,28,11)$
19A3 $19^{2}$ $1332$ $19$ $36$ $( 1,31,19,33, 2, 6, 8,16,17,29,34,32,25,36,21,30,20,38,15)( 3,10,24,12, 4,28, 5,23,13,22, 7,18,11, 9,14,26,27,35,37)$
19A4 $19^{2}$ $1332$ $19$ $36$ $( 1,21,29, 2,15,36,17,33,38,25,16,19,20,32, 8,31,30,34, 6)( 3,14,22, 4,37, 9,13,12,35,11,23,24,27,18, 5,10,26, 7,28)$
19A5 $19^{2}$ $1332$ $19$ $36$ $( 1,17,20, 6,36,19,34,15,16,30, 2,25,31,29,38, 8,21,33,32)( 3,13,27,28, 9,24, 7,37,23,26, 4,11,10,22,35, 5,14,12,18)$
19A6 $19^{2}$ $1332$ $19$ $36$ $( 1,19, 2, 8,17,34,25,21,20,15,31,33, 6,16,29,32,36,30,38)( 3,24, 4, 5,13, 7,11,14,27,37,10,12,28,23,22,18, 9,26,35)$
19A7 $19^{2}$ $1332$ $19$ $36$ $( 1,30,32,16,33,15,21,34, 8,19,38,36,29, 6,31,20,25,17, 2)( 3,26,18,23,12,37,14, 7, 5,24,35, 9,22,28,10,27,11,13, 4)$
19A8 $19^{2}$ $1332$ $19$ $36$ $( 1,29,15,17,38,16,20, 8,30, 6,21, 2,36,33,25,19,32,31,34)( 3,22,37,13,35,23,27, 5,26,28,14, 4, 9,12,11,24,18,10, 7)$
19A9 $19^{2}$ $1332$ $19$ $36$ $( 1,33, 8,29,25,30,15,19, 6,17,32,21,38,31, 2,16,34,36,20)( 3,12, 5,22,11,26,37,24,28,13,18,14,35,10, 4,23, 7, 9,27)$
36A1 $36,1^{2}$ $1406$ $36$ $35$ $( 1,21,18,31,29,12, 5,13,30, 6,15, 2, 3, 8, 7,17,36,16,20,24, 4,23,33,32,37,38,25,34,10,27,35,28,11, 9,22,19)$
36A5 $36,1^{2}$ $1406$ $36$ $35$ $( 1,12,15,17, 4,38,35,19,29, 6, 7,24,37,27,22,31,30, 8,20,32,10, 9,18,13, 3,16,33,34,11,21, 5, 2,36,23,25,28)$
36A7 $36,1^{2}$ $1406$ $36$ $35$ $( 1,13, 7,23,10,19, 5, 8, 4,34,22,12, 3,24,25, 9,29, 2,20,38,11,31,15,16,37,28,18, 6,36,32,35,21,30,17,33,27)$
36A11 $36,1^{2}$ $1406$ $36$ $35$ $( 1, 2,33, 9,30,24,35,12,36,34,18, 8,37,19,15,23,11,13,20,27,29,17,25,21, 3,32,22, 6, 4,28, 5,16,10,31, 7,38)$
36A13 $36,1^{2}$ $1406$ $36$ $35$ $( 1, 8,25,31,36,27, 5,24,11, 6,33,19, 3,38,18,17,10,12,20,28,30,23,22, 2,37,21, 7,34,29,16,35,13, 4, 9,15,32)$
36A17 $36,1^{2}$ $1406$ $36$ $35$ $( 1,16,22,17,11, 8,35, 2,10, 6,25,13,37,12,33,31, 4,21,20,19,36, 9, 7,28, 3,27,15,34,30,38, 5,32,29,23,18,24)$
37A $37,1$ $1368$ $37$ $36$ $( 1,29, 4,17, 7,16,20,31,37,38,10,34,33,18, 5,22,14,11, 6,27,25,21,19, 2,35,32,24, 3,28,13,12,36, 8, 9,15,26,30)$
38A1 $38$ $1332$ $38$ $37$ $( 1,26,36,22,16,12,31,27,21, 7,17, 4,19,35,30,18,29,28,33,37,20,11,34, 5, 2, 3,38, 9,32,23, 6,10,15,14,25,13, 8,24)$
38A3 $38$ $1332$ $38$ $37$ $( 1,22,31, 7,19,18,33,11, 2, 9, 6,14, 8,26,16,27,17,35,29,37,34, 3,32,10,25,24,36,12,21, 4,30,28,20, 5,38,23,15,13)$
38A5 $38$ $1332$ $38$ $37$ $( 1,12,17,18,20, 3, 6,13,36,27,19,28,34, 9,15,24,16, 7,30,37, 2,23,25,26,31, 4,29,11,38,10, 8,22,21,35,33, 5,32,14)$
38A7 $38$ $1332$ $38$ $37$ $( 1,27,30,11,32,13,16, 4,33, 3,15,26,21,18,34,23, 8,12,19,37,38,14,36, 7,29, 5, 6,24,31,35,20, 9,25,22,17,28, 2,10)$
38A9 $38$ $1332$ $38$ $37$ $( 1, 7,33, 9, 8,27,29, 3,25,12,30, 5,15,22,19,11, 6,26,17,37,32,24,21,28,38,13,31,18, 2,14,16,35,34,10,36, 4,20,23)$
38A11 $38$ $1332$ $38$ $37$ $( 1, 4,34,14,31,28,32,26,19, 5,25,27,33,23,36,35, 2,13,21,37, 6,22,30, 3, 8, 7,20,10,16,18,38,24,17,11,15,12,29, 9)$
38A13 $38$ $1332$ $38$ $37$ $( 1,35,38,26,30, 9,36,18,32,22,29,23,16,28, 6,12,33,10,31,37,15,27,20,14,21,11,25, 7,34,13,17, 5, 8, 4, 2,24,19, 3)$
38A15 $38$ $1332$ $38$ $37$ $( 1,18, 6,27,34,24,30,23,31,11, 8,35,32,12,20,13,19, 9,16,37,25, 4,38,22,33,14,17, 3,36,28,15, 7, 2,26,29,10,21, 5)$
38A17 $38$ $1332$ $38$ $37$ $( 1,28,25,35, 6, 7,38,12,34,26,33,13,30,10,17, 9,31, 5,36,37, 8,18,15, 4,32,27, 2,22,20,24,29,14,19,23,21, 3,16,11)$

Malle's constant $a(G)$:     $1/18$

Copy content magma:ConjugacyClasses(G);
 

Character table

39 x 39 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed