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Properties

Label 36T979
36T979 1 18 1->18 33 1->33 2 17 2->17 34 2->34 3 13 3->13 31 3->31 4 14 4->14 32 4->32 5 16 5->16 35 5->35 6 15 6->15 36 6->36 7 7->5 19 7->19 8 8->6 20 8->20 9 9->4 21 9->21 10 10->3 22 10->22 11 11->1 23 11->23 12 12->2 24 12->24 13->12 30 13->30 14->11 29 14->29 15->10 27 15->27 16->9 28 16->28 17->8 25 17->25 18->7 26 18->26 19->4 19->31 20->3 20->32 21->1 21->34 22->2 22->33 23->5 23->36 24->6 24->35 25->11 25->20 26->12 26->19 27->7 27->22 28->8 28->21 29->10 29->23 30->9 30->24 31->14 31->28 32->13 32->27 33->17 33->30 34->18 34->29 35->15 35->25 36->16 36->26
Degree $36$
Order $648$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6^2:C_{18}$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 979);
 

Group invariants

Abstract group:  $C_6^2:C_{18}$
Copy content magma:IdentifyGroup(G);
 
Order:  $648=2^{3} \cdot 3^{4}$
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 $t$:  $979$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $6$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,18,7,5,16,9,4,14,11)(2,17,8,6,15,10,3,13,12)(19,31,28,21,34,29,23,36,26)(20,32,27,22,33,30,24,35,25)$, $(1,33,17,25,11,23,5,35,15,27,7,19,4,32,13,30,9,21)(2,34,18,26,12,24,6,36,16,28,8,20,3,31,14,29,10,22)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$9$:  $C_9$
$12$:  $A_4$
$18$:  $S_3\times C_3$, $C_{18}$
$24$:  $A_4\times C_2$
$36$:  $C_2^2 : C_9$
$54$:  $C_3^2 : C_6$, $C_9\times S_3$
$72$:  12T43, 18T26
$162$:  18T82
$216$:  18T100, 36T182

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: None

Degree 6: $C_6$, $A_4$, $A_4\times C_2$

Degree 9: None

Degree 12: $A_4 \times C_2$

Degree 18: 18T82

Low degree siblings

36T979 x 2

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 $2^{12},1^{12}$ $3$ $2$ $12$ $( 1, 2)( 3, 4)( 5, 6)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)$
2B $2^{18}$ $9$ $2$ $18$ $( 1,27)( 2,28)( 3,26)( 4,25)( 5,30)( 6,29)( 7,33)( 8,34)( 9,35)(10,36)(11,32)(12,31)(13,23)(14,24)(15,21)(16,22)(17,19)(18,20)$
2C $2^{18}$ $27$ $2$ $18$ $( 1,28)( 2,27)( 3,25)( 4,26)( 5,29)( 6,30)( 7,32)( 8,31)( 9,33)(10,34)(11,35)(12,36)(13,20)(14,19)(15,24)(16,23)(17,22)(18,21)$
3A1 $3^{12}$ $1$ $3$ $24$ $( 1, 4, 5)( 2, 3, 6)( 7,11, 9)( 8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,30,27)(26,29,28)(31,36,34)(32,35,33)$
3A-1 $3^{12}$ $1$ $3$ $24$ $( 1, 5, 4)( 2, 6, 3)( 7, 9,11)( 8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,30)(26,28,29)(31,34,36)(32,33,35)$
3B $3^{12}$ $2$ $3$ $24$ $( 1, 4, 5)( 2, 3, 6)( 7,11, 9)( 8,12,10)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,27,30)(26,28,29)(31,34,36)(32,33,35)$
3C1 $3^{6},1^{18}$ $2$ $3$ $12$ $( 1, 5, 4)( 2, 6, 3)( 7, 9,11)( 8,10,12)(13,17,15)(14,18,16)$
3C-1 $3^{6},1^{18}$ $2$ $3$ $12$ $( 1, 4, 5)( 2, 3, 6)( 7,11, 9)( 8,12,10)(13,15,17)(14,16,18)$
3D $3^{8},1^{12}$ $6$ $3$ $16$ $( 1, 4, 5)( 2, 3, 6)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,27,30)(26,28,29)$
3E1 $3^{8},1^{12}$ $6$ $3$ $16$ $( 1, 5, 4)( 2, 6, 3)( 7,11, 9)( 8,12,10)(19,21,23)(20,22,24)(31,36,34)(32,35,33)$
3E-1 $3^{8},1^{12}$ $6$ $3$ $16$ $( 1, 4, 5)( 2, 3, 6)( 7, 9,11)( 8,10,12)(19,23,21)(20,24,22)(31,34,36)(32,33,35)$
6A1 $6^{4},3^{4}$ $3$ $6$ $28$ $( 1, 6, 4, 2, 5, 3)( 7, 9,11)( 8,10,12)(13,18,15,14,17,16)(19,22,23,20,21,24)(25,28,30,26,27,29)(31,34,36)(32,33,35)$
6A-1 $6^{4},3^{4}$ $3$ $6$ $28$ $( 1, 3, 5, 2, 4, 6)( 7,11, 9)( 8,12,10)(13,16,17,14,15,18)(19,24,21,20,23,22)(25,29,27,26,30,28)(31,36,34)(32,35,33)$
6B $6^{4},1^{12}$ $6$ $6$ $20$ $( 1, 6, 4, 2, 5, 3)(13,16,17,14,15,18)(19,22,23,20,21,24)(25,29,27,26,30,28)$
6C $6^{4},3^{4}$ $6$ $6$ $28$ $( 1, 6, 4, 2, 5, 3)( 7,10,11, 8, 9,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,29,27,26,30,28)(31,35,34,32,36,33)$
6D $6^{2},3^{4},2^{6}$ $6$ $6$ $24$ $( 1, 2)( 3, 4)( 5, 6)( 7,11, 9)( 8,12,10)(13,18,15,14,17,16)(19,24,21,20,23,22)(25,26)(27,28)(29,30)(31,34,36)(32,33,35)$
6E $6^{2},3^{4},2^{6}$ $6$ $6$ $24$ $( 1, 2)( 3, 4)( 5, 6)( 7,10,11, 8, 9,12)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,26)(27,28)(29,30)(31,35,34,32,36,33)$
6F1 $6^{2},3^{4},2^{6}$ $6$ $6$ $24$ $( 1, 3, 5, 2, 4, 6)( 7, 9,11)( 8,10,12)(13,14)(15,16)(17,18)(19,24,21,20,23,22)(25,26)(27,28)(29,30)(31,34,36)(32,33,35)$
6F-1 $6^{2},3^{4},2^{6}$ $6$ $6$ $24$ $( 1, 2)( 3, 4)( 5, 6)( 7,11, 9)( 8,12,10)(13,18,15,14,17,16)(19,20)(21,22)(23,24)(25,28,30,26,27,29)(31,36,34)(32,35,33)$
6G1 $6^{2},3^{2},2^{6},1^{6}$ $6$ $6$ $20$ $( 1, 3, 5, 2, 4, 6)( 7,12, 9, 8,11,10)(13,15,17)(14,16,18)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
6G-1 $6^{2},3^{2},2^{6},1^{6}$ $6$ $6$ $20$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(19,21,23)(20,22,24)(25,28,30,26,27,29)(31,33,36,32,34,35)$
6H1 $6^{3},3^{2},2^{3},1^{6}$ $6$ $6$ $22$ $( 1, 6, 4, 2, 5, 3)(13,16,17,14,15,18)(19,20)(21,22)(23,24)(25,28,30,26,27,29)(31,36,34)(32,35,33)$
6H-1 $6^{3},3^{2},2^{3},1^{6}$ $6$ $6$ $22$ $( 1, 3, 5, 2, 4, 6)( 7, 9,11)( 8,10,12)(13,14)(15,16)(17,18)(19,22,23,20,21,24)(25,29,27,26,30,28)$
6I1 $6^{3},3^{2},2^{3},1^{6}$ $6$ $6$ $22$ $( 1, 6, 4, 2, 5, 3)( 7,12, 9, 8,11,10)(19,23,21)(20,24,22)(25,28,30,26,27,29)(31,32)(33,34)(35,36)$
6I-1 $6^{3},3^{2},2^{3},1^{6}$ $6$ $6$ $22$ $( 1, 3, 5, 2, 4, 6)( 7, 8)( 9,10)(11,12)(13,17,15)(14,18,16)(25,29,27,26,30,28)(31,33,36,32,34,35)$
6J1 $6^{6}$ $9$ $6$ $30$ $( 1,25, 5,27, 4,30)( 2,26, 6,28, 3,29)( 7,32, 9,33,11,35)( 8,31,10,34,12,36)(13,21,17,23,15,19)(14,22,18,24,16,20)$
6J-1 $6^{6}$ $9$ $6$ $30$ $( 1,30, 4,27, 5,25)( 2,29, 3,28, 6,26)( 7,35,11,33, 9,32)( 8,36,12,34,10,31)(13,19,15,23,17,21)(14,20,16,24,18,22)$
6K1 $6^{6}$ $27$ $6$ $30$ $( 1,29, 4,28, 5,26)( 2,30, 3,27, 6,25)( 7,33,11,32, 9,35)( 8,34,12,31,10,36)(13,22,15,20,17,24)(14,21,16,19,18,23)$
6K-1 $6^{6}$ $27$ $6$ $30$ $( 1,26, 5,28, 4,29)( 2,25, 6,27, 3,30)( 7,35, 9,32,11,33)( 8,36,10,31,12,34)(13,24,17,20,15,22)(14,23,18,19,16,21)$
9A1 $9^{4}$ $12$ $9$ $32$ $( 1,18,11, 5,16, 7, 4,14, 9)( 2,17,12, 6,15, 8, 3,13,10)(19,31,29,21,34,26,23,36,28)(20,32,30,22,33,25,24,35,27)$
9A-1 $9^{4}$ $12$ $9$ $32$ $( 1, 9,14, 4, 7,16, 5,11,18)( 2,10,13, 3, 8,15, 6,12,17)(19,28,36,23,26,34,21,29,31)(20,27,35,24,25,33,22,30,32)$
9A2 $9^{4}$ $12$ $9$ $32$ $( 1,11,16, 4, 9,18, 5, 7,14)( 2,12,15, 3,10,17, 6, 8,13)(19,29,34,23,28,31,21,26,36)(20,30,33,24,27,32,22,25,35)$
9A-2 $9^{4}$ $12$ $9$ $32$ $( 1,14, 7, 5,18, 9, 4,16,11)( 2,13, 8, 6,17,10, 3,15,12)(19,36,26,21,31,28,23,34,29)(20,35,25,22,32,27,24,33,30)$
9A4 $9^{4}$ $12$ $9$ $32$ $( 1,16, 9, 5,14,11, 4,18, 7)( 2,15,10, 6,13,12, 3,17, 8)(19,34,28,21,36,29,23,31,26)(20,33,27,22,35,30,24,32,25)$
9A-4 $9^{4}$ $12$ $9$ $32$ $( 1, 7,18, 4,11,14, 5, 9,16)( 2, 8,17, 3,12,13, 6,10,15)(19,26,31,23,29,36,21,28,34)(20,25,32,24,30,35,22,27,33)$
9B1 $9^{4}$ $24$ $9$ $32$ $( 1,14,11, 5,18, 7, 4,16, 9)( 2,13,12, 6,17, 8, 3,15,10)(19,31,26,21,34,28,23,36,29)(20,32,25,22,33,27,24,35,30)$
9B-1 $9^{4}$ $24$ $9$ $32$ $( 1, 7,16, 4,11,18, 5, 9,14)( 2, 8,15, 3,12,17, 6,10,13)(19,29,31,23,28,36,21,26,34)(20,30,32,24,27,35,22,25,33)$
9B2 $9^{4}$ $24$ $9$ $32$ $( 1, 9,18, 4, 7,14, 5,11,16)( 2,10,17, 3, 8,13, 6,12,15)(19,26,36,23,29,34,21,28,31)(20,25,35,24,30,33,22,27,32)$
9B-2 $9^{4}$ $24$ $9$ $32$ $( 1,16, 7, 5,14, 9, 4,18,11)( 2,15, 8, 6,13,10, 3,17,12)(19,36,28,21,31,29,23,34,26)(20,35,27,22,32,30,24,33,25)$
9B4 $9^{4}$ $24$ $9$ $32$ $( 1,18, 9, 5,16,11, 4,14, 7)( 2,17,10, 6,15,12, 3,13, 8)(19,34,29,21,36,26,23,31,28)(20,33,30,22,35,25,24,32,27)$
9B-4 $9^{4}$ $24$ $9$ $32$ $( 1,11,14, 4, 9,16, 5, 7,18)( 2,12,13, 3,10,15, 6, 8,17)(19,28,34,23,26,31,21,29,36)(20,27,33,24,25,32,22,30,35)$
18A1 $18^{2}$ $36$ $18$ $34$ $( 1,33,18,25,11,24, 5,35,16,27, 7,20, 4,32,14,30, 9,22)( 2,34,17,26,12,23, 6,36,15,28, 8,19, 3,31,13,29,10,21)$
18A-1 $18^{2}$ $36$ $18$ $34$ $( 1,22, 9,30,14,32, 4,20, 7,27,16,35, 5,24,11,25,18,33)( 2,21,10,29,13,31, 3,19, 8,28,15,36, 6,23,12,26,17,34)$
18A5 $18^{2}$ $36$ $18$ $34$ $( 1,24, 7,30,18,35, 4,22,11,27,14,33, 5,20, 9,25,16,32)( 2,23, 8,29,17,36, 3,21,12,28,13,34, 6,19,10,26,15,31)$
18A-5 $18^{2}$ $36$ $18$ $34$ $( 1,32,16,25, 9,20, 5,33,14,27,11,22, 4,35,18,30, 7,24)( 2,31,15,26,10,19, 6,34,13,28,12,21, 3,36,17,29, 8,23)$
18A7 $18^{2}$ $36$ $18$ $34$ $( 1,35,14,25, 7,22, 5,32,18,27, 9,24, 4,33,16,30,11,20)( 2,36,13,26, 8,21, 6,31,17,28,10,23, 3,34,15,29,12,19)$
18A-7 $18^{2}$ $36$ $18$ $34$ $( 1,20,11,30,16,33, 4,24, 9,27,18,32, 5,22, 7,25,14,35)( 2,19,12,29,15,34, 3,23,10,28,17,31, 6,21, 8,26,13,36)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

48 x 48 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed