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Properties

Label 36T40
36T40 1 9 1->9 34 1->34 35 1->35 2 10 2->10 33 2->33 36 2->36 3 12 3->12 3->34 3->35 4 11 4->11 4->33 4->36 5 5->3 6 5->6 17 5->17 6->4 18 6->18 7 7->2 8 7->8 19 7->19 8->1 20 8->20 14 9->14 29 9->29 13 10->13 30 10->30 15 11->15 31 11->31 16 12->16 32 12->32 22 13->22 13->33 21 14->21 14->34 23 15->23 15->36 24 16->24 16->35 17->18 25 17->25 26 18->26 19->20 28 19->28 27 20->27 21->17 21->26 21->28 22->18 22->25 22->27 23->19 23->25 23->27 24->20 24->26 24->28 25->10 26->9 27->11 28->12 29->13 29->30 29->31 30->14 30->32 31->16 31->32 32->15 33->8 34->7 35->6 36->5
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times D_6$

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Copy content magma:G := TransitiveGroup(36, 40);
 

Group invariants

Abstract group:  $S_3\times D_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $72=2^{3} \cdot 3^{2}$
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 $n$:  $36$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $40$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $12$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,35,6,4,33,8)(2,36,5,3,34,7)(9,29,13,22,18,26)(10,30,14,21,17,25)(11,31,16,24,20,27)(12,32,15,23,19,28)$, $(1,34)(2,33)(3,35)(4,36)(5,6)(7,8)(9,14)(10,13)(11,15)(12,16)(17,18)(19,20)(21,26)(22,25)(23,27)(24,28)(29,30)(31,32)$, $(1,9)(2,10)(3,12)(4,11)(5,17)(6,18)(7,19)(8,20)(13,33)(14,34)(15,36)(16,35)(21,28)(22,27)(23,25)(24,26)(29,31)(30,32)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$ x 2
$8$:  $C_2^3$
$12$:  $D_{6}$ x 6
$24$:  $S_3 \times C_2^2$ x 2
$36$:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$ x 2

Degree 4: $C_2^2$

Degree 6: $S_3$, $D_{6}$ x 5

Degree 9: $S_3^2$

Degree 12: $D_6$, $S_3 \times C_2^2$

Degree 18: $S_3^2$, 18T29 x 2

Low degree siblings

12T37 x 2, 18T29 x 4, 24T73, 36T34 x 2, 36T40 x 3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{36}$ $1$ $1$ $0$ $()$
2A $2^{18}$ $1$ $2$ $18$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
2B $2^{18}$ $3$ $2$ $18$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,22)(10,21)(11,24)(12,23)(13,26)(14,25)(15,28)(16,27)(17,30)(18,29)(19,32)(20,31)(33,35)(34,36)$
2C $2^{18}$ $3$ $2$ $18$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,18)( 6,17)( 7,20)( 8,19)( 9,34)(10,33)(11,36)(12,35)(21,24)(22,23)(25,27)(26,28)(29,32)(30,31)$
2D $2^{18}$ $3$ $2$ $18$ $( 1,34)( 2,33)( 3,35)( 4,36)( 5, 6)( 7, 8)( 9,14)(10,13)(11,15)(12,16)(17,18)(19,20)(21,26)(22,25)(23,27)(24,28)(29,30)(31,32)$
2E $2^{12},1^{12}$ $3$ $2$ $12$ $( 1,33)( 2,34)( 3,36)( 4,35)( 9,13)(10,14)(11,16)(12,15)(21,25)(22,26)(23,28)(24,27)$
2F $2^{18}$ $9$ $2$ $18$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,22)(10,21)(11,24)(12,23)(13,29)(14,30)(15,32)(16,31)(17,25)(18,26)(19,28)(20,27)(33,35)(34,36)$
2G $2^{18}$ $9$ $2$ $18$ $( 1,32)( 2,31)( 3,29)( 4,30)( 5,27)( 6,28)( 7,26)( 8,25)( 9,12)(10,11)(13,19)(14,20)(15,18)(16,17)(21,35)(22,36)(23,33)(24,34)$
3A $3^{12}$ $2$ $3$ $24$ $( 1,26,16)( 2,25,15)( 3,28,14)( 4,27,13)( 5,30,19)( 6,29,20)( 7,32,17)( 8,31,18)( 9,35,24)(10,36,23)(11,33,22)(12,34,21)$
3B $3^{12}$ $2$ $3$ $24$ $( 1,33, 6)( 2,34, 5)( 3,36, 7)( 4,35, 8)( 9,18,13)(10,17,14)(11,20,16)(12,19,15)(21,30,25)(22,29,26)(23,32,28)(24,31,27)$
3C $3^{12}$ $4$ $3$ $24$ $( 1,22,20)( 2,21,19)( 3,23,17)( 4,24,18)( 5,25,12)( 6,26,11)( 7,28,10)( 8,27, 9)(13,35,31)(14,36,32)(15,34,30)(16,33,29)$
6A $6^{6}$ $2$ $6$ $30$ $( 1,15,26, 2,16,25)( 3,13,28, 4,14,27)( 5,20,30, 6,19,29)( 7,18,32, 8,17,31)( 9,23,35,10,24,36)(11,21,33,12,22,34)$
6B $6^{6}$ $2$ $6$ $30$ $( 1, 5,33, 2, 6,34)( 3, 8,36, 4, 7,35)( 9,14,18,10,13,17)(11,15,20,12,16,19)(21,26,30,22,25,29)(23,27,32,24,28,31)$
6C $6^{6}$ $4$ $6$ $30$ $( 1,19,22, 2,20,21)( 3,18,23, 4,17,24)( 5,11,25, 6,12,26)( 7, 9,28, 8,10,27)(13,32,35,14,31,36)(15,29,34,16,30,33)$
6D $6^{6}$ $6$ $6$ $30$ $( 1,35, 6, 4,33, 8)( 2,36, 5, 3,34, 7)( 9,29,13,22,18,26)(10,30,14,21,17,25)(11,31,16,24,20,27)(12,32,15,23,19,28)$
6E $6^{6}$ $6$ $6$ $30$ $( 1,10, 6,14,33,17)( 2, 9, 5,13,34,18)( 3,11, 7,16,36,20)( 4,12, 8,15,35,19)(21,31,25,24,30,27)(22,32,26,23,29,28)$
6F $6^{6}$ $6$ $6$ $30$ $( 1,21,16,34,26,12)( 2,22,15,33,25,11)( 3,24,14,35,28, 9)( 4,23,13,36,27,10)( 5,29,19, 6,30,20)( 7,31,17, 8,32,18)$
6G $6^{4},3^{4}$ $6$ $6$ $28$ $( 1,11,26,33,16,22)( 2,12,25,34,15,21)( 3,10,28,36,14,23)( 4, 9,27,35,13,24)( 5,19,30)( 6,20,29)( 7,17,32)( 8,18,31)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 6A 6B 6C 6D 6E 6F 6G
Size 1 1 3 3 3 3 9 9 2 2 4 2 2 4 6 6 6 6
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A 3B 3C 3A 3B 3C 3B 3B 3A 3A
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 1A 1A 2A 2A 2A 2B 2C 2D 2E
Type
72.46.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.46.1b R 1 −1 −1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 −1 1
72.46.1c R 1 −1 −1 1 1 −1 1 −1 1 1 1 −1 −1 −1 −1 1 1 −1
72.46.1d R 1 −1 1 −1 −1 1 1 −1 1 1 1 −1 −1 −1 1 −1 −1 1
72.46.1e R 1 −1 1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 1 −1 1 −1
72.46.1f R 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
72.46.1g R 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 1 1
72.46.1h R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 −1 −1
72.46.2a R 2 2 0 0 2 2 0 0 −1 2 −1 −1 2 −1 0 0 −1 −1
72.46.2b R 2 2 2 2 0 0 0 0 2 −1 −1 2 −1 −1 −1 −1 0 0
72.46.2c R 2 −2 −2 2 0 0 0 0 2 −1 −1 −2 1 1 1 −1 0 0
72.46.2d R 2 −2 0 0 −2 2 0 0 −1 2 −1 1 −2 1 0 0 1 −1
72.46.2e R 2 −2 0 0 2 −2 0 0 −1 2 −1 1 −2 1 0 0 −1 1
72.46.2f R 2 −2 2 −2 0 0 0 0 2 −1 −1 −2 1 1 −1 1 0 0
72.46.2g R 2 2 −2 −2 0 0 0 0 2 −1 −1 2 −1 −1 1 1 0 0
72.46.2h R 2 2 0 0 −2 −2 0 0 −1 2 −1 −1 2 −1 0 0 1 1
72.46.4a R 4 4 0 0 0 0 0 0 −2 −2 1 −2 −2 1 0 0 0 0
72.46.4b R 4 −4 0 0 0 0 0 0 −2 −2 1 2 2 −1 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed