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Properties

Label 37T9
37T9 1 2 1->2 1->2 3 2->3 4 2->4 3->4 6 3->6 5 4->5 8 4->8 5->6 10 5->10 7 6->7 12 6->12 7->8 14 7->14 9 8->9 16 8->16 9->10 18 9->18 11 10->11 20 10->20 11->12 22 11->22 13 12->13 24 12->24 13->14 26 13->26 15 14->15 28 14->28 15->16 30 15->30 17 16->17 32 16->32 17->18 34 17->34 19 18->19 36 18->36 19->1 19->20 20->3 21 20->21 21->5 21->22 22->7 23 22->23 23->9 23->24 24->11 25 24->25 25->13 25->26 26->15 27 26->27 27->17 27->28 28->19 29 28->29 29->21 29->30 30->23 31 30->31 31->25 31->32 32->27 33 32->33 33->29 33->34 34->31 35 34->35 35->33 35->36 36->35 37 36->37 37->1
Degree $37$
Order $1332$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $F_{37}$

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Copy content magma:G := TransitiveGroup(37, 9);
 

Group invariants

Abstract group:  $F_{37}$
Copy content magma:IdentifyGroup(G);
 
Order:  $1332=2^{2} \cdot 3^{2} \cdot 37$
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 $n$:  $37$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9$
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:  $(1,2,4,8,16,32,27,17,34,31,25,13,26,15,30,23,9,18,36,35,33,29,21,5,10,20,3,6,12,24,11,22,7,14,28,19)$, $(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$9$:  $C_9$
$12$:  $C_{12}$
$18$:  $C_{18}$
$36$:  $C_{36}$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - 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 $1^{37}$ $1$ $1$ $0$ $()$
2A $2^{18},1$ $37$ $2$ $18$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)(17,37)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)$
3A1 $3^{12},1$ $37$ $3$ $24$ $( 1,26,17)( 2,36, 6)( 3, 9,32)( 4,19,21)( 5,29,10)( 7,12,25)( 8,22,14)(11,15,18)(13,35,33)(16,28,37)(20,31,30)(23,24,34)$
3A-1 $3^{12},1$ $37$ $3$ $24$ $( 1,17,26)( 2, 6,36)( 3,32, 9)( 4,21,19)( 5,10,29)( 7,25,12)( 8,14,22)(11,18,15)(13,33,35)(16,37,28)(20,30,31)(23,34,24)$
4A1 $4^{9},1$ $37$ $4$ $27$ $( 1,35,16,19)( 2,29,15,25)( 3,23,14,31)( 4,17,13,37)( 5,11,12, 6)( 7,36,10,18)( 8,30, 9,24)(20,32,34,22)(21,26,33,28)$
4A-1 $4^{9},1$ $37$ $4$ $27$ $( 1,19,16,35)( 2,25,15,29)( 3,31,14,23)( 4,37,13,17)( 5, 6,12,11)( 7,18,10,36)( 8,24, 9,30)(20,22,34,32)(21,28,33,26)$
6A1 $6^{6},1$ $37$ $6$ $30$ $( 1,37,26,16,17,28)( 2,11,36,15, 6,18)( 3,22, 9,14,32, 8)( 4,33,19,13,21,35)( 5, 7,29,12,10,25)(20,24,31,34,30,23)$
6A-1 $6^{6},1$ $37$ $6$ $30$ $( 1,28,17,16,26,37)( 2,18, 6,15,36,11)( 3, 8,32,14, 9,22)( 4,35,21,13,19,33)( 5,25,10,12,29, 7)(20,23,30,34,31,24)$
9A1 $9^{4},1$ $37$ $9$ $32$ $( 1,30,11,26,20,15,17,31,18)( 2,37,23,36,16,24, 6,28,34)( 3, 7,35, 9,12,33,32,25,13)( 4,14,10,19, 8, 5,21,22,29)$
9A-1 $9^{4},1$ $37$ $9$ $32$ $( 1,18,31,17,15,20,26,11,30)( 2,34,28, 6,24,16,36,23,37)( 3,13,25,32,33,12, 9,35, 7)( 4,29,22,21, 5, 8,19,10,14)$
9A2 $9^{4},1$ $37$ $9$ $32$ $( 1,11,20,17,18,30,26,15,31)( 2,23,16, 6,34,37,36,24,28)( 3,35,12,32,13, 7, 9,33,25)( 4,10, 8,21,29,14,19, 5,22)$
9A-2 $9^{4},1$ $37$ $9$ $32$ $( 1,31,15,26,30,18,17,20,11)( 2,28,24,36,37,34, 6,16,23)( 3,25,33, 9, 7,13,32,12,35)( 4,22, 5,19,14,29,21, 8,10)$
9A4 $9^{4},1$ $37$ $9$ $32$ $( 1,20,18,26,31,11,17,30,15)( 2,16,34,36,28,23, 6,37,24)( 3,12,13, 9,25,35,32, 7,33)( 4, 8,29,19,22,10,21,14, 5)$
9A-4 $9^{4},1$ $37$ $9$ $32$ $( 1,15,30,17,11,31,26,18,20)( 2,24,37, 6,23,28,36,34,16)( 3,33, 7,32,35,25, 9,13,12)( 4, 5,14,21,10,22,19,29, 8)$
12A1 $12^{3},1$ $37$ $12$ $33$ $( 1,21,37,35,26, 4,16,33,17,19,28,13)( 2, 7,11,29,36,12,15,10, 6,25,18, 5)( 3,30,22,23, 9,20,14,24,32,31, 8,34)$
12A-1 $12^{3},1$ $37$ $12$ $33$ $( 1,13,28,19,17,33,16, 4,26,35,37,21)( 2, 5,18,25, 6,10,15,12,36,29,11, 7)( 3,34, 8,31,32,24,14,20, 9,23,22,30)$
12A5 $12^{3},1$ $37$ $12$ $33$ $( 1, 4,28,35,17,21,16,13,26,19,37,33)( 2,12,18,29, 6, 7,15, 5,36,25,11,10)( 3,20, 8,23,32,30,14,34, 9,31,22,24)$
12A-5 $12^{3},1$ $37$ $12$ $33$ $( 1,33,37,19,26,13,16,21,17,35,28, 4)( 2,10,11,25,36, 5,15, 7, 6,29,18,12)( 3,24,22,31, 9,34,14,30,32,23, 8,20)$
18A1 $18^{2},1$ $37$ $18$ $34$ $( 1, 2,30,37,11,23,26,36,20,16,15,24,17, 6,31,28,18,34)( 3,21, 7,22,35,29, 9, 4,12,14,33,10,32,19,25, 8,13, 5)$
18A-1 $18^{2},1$ $37$ $18$ $34$ $( 1,34,18,28,31, 6,17,24,15,16,20,36,26,23,11,37,30, 2)( 3, 5,13, 8,25,19,32,10,33,14,12, 4, 9,29,35,22, 7,21)$
18A5 $18^{2},1$ $37$ $18$ $34$ $( 1,23,15,28,30,36,17,34,11,16,31, 2,26,24,18,37,20, 6)( 3,29,33, 8, 7, 4,32, 5,35,14,25,21, 9,10,13,22,12,19)$
18A-5 $18^{2},1$ $37$ $18$ $34$ $( 1, 6,20,37,18,24,26, 2,31,16,11,34,17,36,30,28,15,23)( 3,19,12,22,13,10, 9,21,25,14,35, 5,32, 4, 7, 8,33,29)$
18A7 $18^{2},1$ $37$ $18$ $34$ $( 1,36,31,37,15,34,26, 6,30,16,18,23,17, 2,20,28,11,24)( 3, 4,25,22,33, 5, 9,19, 7,14,13,29,32,21,12, 8,35,10)$
18A-7 $18^{2},1$ $37$ $18$ $34$ $( 1,24,11,28,20, 2,17,23,18,16,30, 6,26,34,15,37,31,36)( 3,10,35, 8,12,21,32,29,13,14, 7,19, 9, 5,33,22,25, 4)$
36A1 $36,1$ $37$ $36$ $35$ $( 1, 3, 2,21,30, 7,37,22,11,35,23,29,26, 9,36, 4,20,12,16,14,15,33,24,10,17,32, 6,19,31,25,28, 8,18,13,34, 5)$
36A-1 $36,1$ $37$ $36$ $35$ $( 1, 5,34,13,18, 8,28,25,31,19, 6,32,17,10,24,33,15,14,16,12,20, 4,36, 9,26,29,23,35,11,22,37, 7,30,21, 2, 3)$
36A5 $36,1$ $37$ $36$ $35$ $( 1, 7,23, 4,15,32,28, 5,30,35,36,14,17,25,34,21,11, 9,16,10,31,13, 2,22,26,12,24,19,18, 3,37,29,20,33, 6, 8)$
36A-5 $36,1$ $37$ $36$ $35$ $( 1, 8, 6,33,20,29,37, 3,18,19,24,12,26,22, 2,13,31,10,16, 9,11,21,34,25,17,14,36,35,30, 5,28,32,15, 4,23, 7)$
36A7 $36,1$ $37$ $36$ $35$ $( 1,22,36,33,31, 5,37, 9,15,19,34, 7,26,14, 6,13,30,29,16,32,18,21,23,12,17, 8, 2,35,20,10,28, 3,11, 4,24,25)$
36A-7 $36,1$ $37$ $36$ $35$ $( 1,25,24, 4,11, 3,28,10,20,35, 2, 8,17,12,23,21,18,32,16,29,30,13, 6,14,26, 7,34,19,15, 9,37, 5,31,33,36,22)$
36A11 $36,1$ $37$ $36$ $35$ $( 1,29,24,13,11,14,28, 7,20,19, 2, 9,17, 5,23,33,18,22,16,25,30, 4, 6, 3,26,10,34,35,15, 8,37,12,31,21,36,32)$
36A-11 $36,1$ $37$ $36$ $35$ $( 1,32,36,21,31,12,37, 8,15,35,34,10,26, 3, 6, 4,30,25,16,22,18,33,23, 5,17, 9, 2,19,20, 7,28,14,11,13,24,29)$
36A13 $36,1$ $37$ $36$ $35$ $( 1, 9, 6,21,20,25,37,14,18,35,24, 5,26,32, 2, 4,31, 7,16, 8,11,33,34,29,17, 3,36,19,30,12,28,22,15,13,23,10)$
36A-13 $36,1$ $37$ $36$ $35$ $( 1,10,23,13,15,22,28,12,30,19,36, 3,17,29,34,33,11, 8,16, 7,31, 4, 2,32,26, 5,24,35,18,14,37,25,20,21, 6, 9)$
36A17 $36,1$ $37$ $36$ $35$ $( 1,12,34, 4,18, 9,28,29,31,35, 6,22,17, 7,24,21,15, 3,16, 5,20,13,36, 8,26,25,23,19,11,32,37,10,30,33, 2,14)$
36A-17 $36,1$ $37$ $36$ $35$ $( 1,14, 2,33,30,10,37,32,11,19,23,25,26, 8,36,13,20, 5,16, 3,15,21,24, 7,17,22, 6,35,31,29,28, 9,18, 4,34,12)$
37A $37$ $36$ $37$ $36$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)$

Malle's constant $a(G)$:     $1/18$

Copy content magma:ConjugacyClasses(G);
 

Character table

37 x 37 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed