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Properties

Label 36T497
36T497 1 29 1->29 34 1->34 2 30 2->30 33 2->33 3 25 3->25 36 3->36 4 26 4->26 35 4->35 5 27 5->27 31 5->31 6 28 6->28 32 6->32 7 23 7->23 7->27 8 24 8->24 8->28 9 19 9->19 9->30 10 20 10->20 10->29 11 22 11->22 11->26 12 21 12->21 12->25 13 13->22 13->35 14 14->21 14->36 15 15->24 15->32 16 16->23 16->31 17 17->20 17->34 18 18->19 18->33 19->3 19->3 20->4 20->4 21->6 21->6 22->5 22->5 23->2 23->2 24->1 24->1 25->9 25->9 26->10 26->10 27->11 27->11 28->12 28->12 29->8 29->8 30->7 30->7 31->15 31->15 32->16 32->16 33->17 33->17 34->18 34->18 35->14 35->14 36->13 36->13
Degree $36$
Order $324$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\wr C_4$

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Copy content magma:G := TransitiveGroup(36, 497);
 

Group invariants

Abstract group:  $C_3\wr C_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $324=2^{2} \cdot 3^{4}$
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 $t$:  $497$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $18$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,34,18,19,3,36,13,22,5,31,15,24)(2,33,17,20,4,35,14,21,6,32,16,23)(7,27,11,26,10,29,8,28,12,25,9,30)$, $(1,29,8,24)(2,30,7,23)(3,25,9,19)(4,26,10,20)(5,27,11,22)(6,28,12,21)(13,35,14,36)(15,32,16,31)(17,34,18,33)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $S_3$, $C_6$
$12$:  $C_{12}$, $C_3 : C_4$
$18$:  $S_3\times C_3$
$36$:  $C_3^2:C_4$, $C_3\times (C_3 : C_4)$
$108$:  12T72, 12T73

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: $S_3\times C_3$, $C_3^2:C_4$

Degree 9: None

Degree 12: $(C_3\times C_3):C_4$, $C_3\times (C_3 : C_4)$, 12T131

Degree 18: 18T123

Low degree siblings

12T131 x 4, 18T123 x 4, 36T497 x 3, 36T514 x 2, 36T527 x 2, 36T532 x 4, 36T536 x 4, 36T543 x 4, 36T544 x 8

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 $2^{18}$ $9$ $2$ $18$ $( 1, 8)( 2, 7)( 3, 9)( 4,10)( 5,11)( 6,12)(13,14)(15,16)(17,18)(19,25)(20,26)(21,28)(22,27)(23,30)(24,29)(31,32)(33,34)(35,36)$
3A1 $3^{12}$ $1$ $3$ $24$ $( 1, 5, 3)( 2, 6, 4)( 7,12,10)( 8,11, 9)(13,18,15)(14,17,16)(19,24,22)(20,23,21)(25,29,27)(26,30,28)(31,36,34)(32,35,33)$
3A-1 $3^{12}$ $1$ $3$ $24$ $( 1, 3, 5)( 2, 4, 6)( 7,10,12)( 8, 9,11)(13,15,18)(14,16,17)(19,22,24)(20,21,23)(25,27,29)(26,28,30)(31,34,36)(32,33,35)$
3B $3^{12}$ $2$ $3$ $24$ $( 1, 5, 3)( 2, 6, 4)( 7,12,10)( 8,11, 9)(13,18,15)(14,17,16)(19,22,24)(20,21,23)(25,27,29)(26,28,30)(31,34,36)(32,33,35)$
3C1 $3^{6},1^{18}$ $2$ $3$ $12$ $(19,24,22)(20,23,21)(25,29,27)(26,30,28)(31,36,34)(32,35,33)$
3C-1 $3^{6},1^{18}$ $2$ $3$ $12$ $(19,22,24)(20,21,23)(25,27,29)(26,28,30)(31,34,36)(32,33,35)$
3D $3^{12}$ $4$ $3$ $24$ $( 1, 7,14)( 2, 8,13)( 3,10,16)( 4, 9,15)( 5,12,17)( 6,11,18)(19,26,32)(20,25,31)(21,27,34)(22,28,33)(23,29,36)(24,30,35)$
3E $3^{6},1^{18}$ $4$ $3$ $12$ $(19,32,26)(20,31,25)(21,34,27)(22,33,28)(23,36,29)(24,35,30)$
3F1 $3^{12}$ $4$ $3$ $24$ $( 1,10,17)( 2, 9,18)( 3,12,14)( 4,11,13)( 5, 7,16)( 6, 8,15)(19,30,33)(20,29,34)(21,25,36)(22,26,35)(23,27,31)(24,28,32)$
3F-1 $3^{12}$ $4$ $3$ $24$ $( 1,16,12)( 2,15,11)( 3,17, 7)( 4,18, 8)( 5,14,10)( 6,13, 9)(19,30,33)(20,29,34)(21,25,36)(22,26,35)(23,27,31)(24,28,32)$
3G1 $3^{12}$ $4$ $3$ $24$ $( 1,10,17)( 2, 9,18)( 3,12,14)( 4,11,13)( 5, 7,16)( 6, 8,15)(19,28,35)(20,27,36)(21,29,31)(22,30,32)(23,25,34)(24,26,33)$
3G-1 $3^{12}$ $4$ $3$ $24$ $( 1,12,16)( 2,11,15)( 3, 7,17)( 4, 8,18)( 5,10,14)( 6, 9,13)(19,30,33)(20,29,34)(21,25,36)(22,26,35)(23,27,31)(24,28,32)$
3H1 $3^{12}$ $4$ $3$ $24$ $( 1, 3, 5)( 2, 4, 6)( 7,10,12)( 8, 9,11)(13,15,18)(14,16,17)(19,35,28)(20,36,27)(21,31,29)(22,32,30)(23,34,25)(24,33,26)$
3H-1 $3^{12}$ $4$ $3$ $24$ $( 1,10,17)( 2, 9,18)( 3,12,14)( 4,11,13)( 5, 7,16)( 6, 8,15)(19,24,22)(20,23,21)(25,29,27)(26,30,28)(31,36,34)(32,35,33)$
3I1 $3^{12}$ $4$ $3$ $24$ $( 1, 3, 5)( 2, 4, 6)( 7,10,12)( 8, 9,11)(13,15,18)(14,16,17)(19,33,30)(20,34,29)(21,36,25)(22,35,26)(23,31,27)(24,32,28)$
3I-1 $3^{12}$ $4$ $3$ $24$ $( 1, 5, 3)( 2, 6, 4)( 7,12,10)( 8,11, 9)(13,18,15)(14,17,16)(19,35,28)(20,36,27)(21,31,29)(22,32,30)(23,34,25)(24,33,26)$
3J1 $3^{12}$ $4$ $3$ $24$ $( 1,12,16)( 2,11,15)( 3, 7,17)( 4, 8,18)( 5,10,14)( 6, 9,13)(19,26,32)(20,25,31)(21,27,34)(22,28,33)(23,29,36)(24,30,35)$
3J-1 $3^{12}$ $4$ $3$ $24$ $( 1,14, 7)( 2,13, 8)( 3,16,10)( 4,15, 9)( 5,17,12)( 6,18,11)(19,28,35)(20,27,36)(21,29,31)(22,30,32)(23,25,34)(24,26,33)$
3K1 $3^{12}$ $4$ $3$ $24$ $( 1,17,10)( 2,18, 9)( 3,14,12)( 4,13,11)( 5,16, 7)( 6,15, 8)(19,26,32)(20,25,31)(21,27,34)(22,28,33)(23,29,36)(24,30,35)$
3K-1 $3^{12}$ $4$ $3$ $24$ $( 1, 7,14)( 2, 8,13)( 3,10,16)( 4, 9,15)( 5,12,17)( 6,11,18)(19,28,35)(20,27,36)(21,29,31)(22,30,32)(23,25,34)(24,26,33)$
3L1 $3^{12}$ $4$ $3$ $24$ $( 1, 5, 3)( 2, 6, 4)( 7,12,10)( 8,11, 9)(13,18,15)(14,17,16)(19,32,26)(20,31,25)(21,34,27)(22,33,28)(23,36,29)(24,35,30)$
3L-1 $3^{12}$ $4$ $3$ $24$ $( 1, 7,14)( 2, 8,13)( 3,10,16)( 4, 9,15)( 5,12,17)( 6,11,18)(19,22,24)(20,21,23)(25,27,29)(26,28,30)(31,34,36)(32,33,35)$
3M1 $3^{6},1^{18}$ $4$ $3$ $12$ $( 1,12,16)( 2,11,15)( 3, 7,17)( 4, 8,18)( 5,10,14)( 6, 9,13)$
3M-1 $3^{6},1^{18}$ $4$ $3$ $12$ $(19,33,30)(20,34,29)(21,36,25)(22,35,26)(23,31,27)(24,32,28)$
4A1 $4^{9}$ $27$ $4$ $27$ $( 1,26, 2,25)( 3,28, 4,27)( 5,30, 6,29)( 7,32,13,20)( 8,31,14,19)( 9,34,16,22)(10,33,15,21)(11,36,17,24)(12,35,18,23)$
4A-1 $4^{9}$ $27$ $4$ $27$ $( 1,25, 2,26)( 3,27, 4,28)( 5,29, 6,30)( 7,20,13,32)( 8,19,14,31)( 9,22,16,34)(10,21,15,33)(11,24,17,36)(12,23,18,35)$
6A1 $6^{6}$ $9$ $6$ $30$ $( 1, 4, 5, 2, 3, 6)( 7,15,12,13,10,18)( 8,16,11,14, 9,17)(19,34,24,31,22,36)(20,33,23,32,21,35)(25,28,29,26,27,30)$
6A-1 $6^{6}$ $9$ $6$ $30$ $( 1, 6, 3, 2, 5, 4)( 7,18,10,13,12,15)( 8,17, 9,14,11,16)(19,36,22,31,24,34)(20,35,21,32,23,33)(25,30,27,26,29,28)$
6B $6^{6}$ $18$ $6$ $30$ $( 1, 9, 5, 8, 3,11)( 2,10, 6, 7, 4,12)(13,16,18,14,15,17)(19,29,22,25,24,27)(20,30,21,26,23,28)(31,35,34,32,36,33)$
6C1 $6^{3},2^{9}$ $18$ $6$ $24$ $( 1, 8)( 2, 7)( 3, 9)( 4,10)( 5,11)( 6,12)(13,14)(15,16)(17,18)(19,27,24,25,22,29)(20,28,23,26,21,30)(31,33,36,32,34,35)$
6C-1 $6^{3},2^{9}$ $18$ $6$ $24$ $( 1,11, 3, 8, 5, 9)( 2,12, 4, 7, 6,10)(13,17,15,14,18,16)(19,25)(20,26)(21,28)(22,27)(23,30)(24,29)(31,32)(33,34)(35,36)$
12A1 $12^{3}$ $27$ $12$ $33$ $( 1,29, 4,26, 5,27, 2,30, 3,25, 6,28)( 7,23,15,32,12,21,13,35,10,20,18,33)( 8,24,16,31,11,22,14,36, 9,19,17,34)$
12A-1 $12^{3}$ $27$ $12$ $33$ $( 1,28, 6,25, 3,30, 2,27, 5,26, 4,29)( 7,33,18,20,10,35,13,21,12,32,15,23)( 8,34,17,19, 9,36,14,22,11,31,16,24)$
12A5 $12^{3}$ $27$ $12$ $33$ $( 1,27, 6,26, 3,29, 2,28, 5,25, 4,30)( 7,21,18,32,10,23,13,33,12,20,15,35)( 8,22,17,31, 9,24,14,34,11,19,16,36)$
12A-5 $12^{3}$ $27$ $12$ $33$ $( 1,30, 4,25, 5,28, 2,29, 3,26, 6,27)( 7,35,15,20,12,33,13,23,10,32,18,21)( 8,36,16,19,11,34,14,24, 9,31,17,22)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

36 x 36 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed