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Properties

Label 36T528
36T528 1 3 1->3 8 1->8 22 1->22 2 2->1 7 2->7 23 2->23 3->2 9 3->9 24 3->24 4 4->23 32 4->32 5 5->24 33 5->33 6 6->22 31 6->31 7->5 8->4 9->6 10 10->1 11 10->11 29 10->29 11->2 12 11->12 28 11->28 12->3 12->10 30 12->30 13 15 13->15 19 13->19 35 13->35 14 14->13 21 14->21 34 14->34 15->14 20 15->20 36 15->36 16 16->8 16->36 17 17->9 17->34 18 18->7 18->35 19->16 20->17 21->18 22->15 22->23 23->13 23->24 24->14 24->22 25 25->12 26 25->26 25->32 26->11 27 26->27 26->31 27->10 27->25 27->33 28->20 29->21 30->19 31->28 32->30 33->29 34->25 34->36 35->26 35->34 36->27 36->35
Degree $36$
Order $324$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2:S_3^2$

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Copy content magma:G := TransitiveGroup(36, 528);
 

Group invariants

Abstract group:  $C_3^2:S_3^2$
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 $t$:  $528$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $3$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,22,15,36,27,10)(2,23,13,35,26,11)(3,24,14,34,25,12)(4,32,30,19,16,8)(5,33,29,21,18,7)(6,31,28,20,17,9)$, $(1,8)(2,7)(3,9)(4,23)(5,24)(6,22)(10,29)(11,28)(12,30)(13,19)(14,21)(15,20)(16,36)(17,34)(18,35)(25,32)(26,31)(27,33)$, $(1,3,2)(10,11,12)(13,15,14)(22,23,24)(25,26,27)(34,36,35)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 5
$12$:  $D_{6}$ x 5
$18$:  $C_3^2:C_2$
$36$:  $S_3^2$ x 4, 18T12
$108$:  $C_3^2 : D_{6} $, 18T58

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $S_3^2$

Degree 9: None

Degree 12: $S_3^2$

Degree 18: None

Low degree siblings

18T133, 18T139, 27T117 x 3, 27T127, 36T506, 36T509

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 $2^{18}$ $9$ $2$ $18$ $( 1,29)( 2,28)( 3,30)( 4,14)( 5,15)( 6,13)( 7,36)( 8,34)( 9,35)(10,21)(11,20)(12,19)(16,25)(17,26)(18,27)(22,33)(23,31)(24,32)$
2B $2^{18}$ $27$ $2$ $18$ $( 1,24)( 2,22)( 3,23)( 4, 9)( 5, 8)( 6, 7)(10,27)(11,26)(12,25)(13,34)(14,36)(15,35)(16,21)(17,19)(18,20)(28,31)(29,33)(30,32)$
2C $2^{18}$ $27$ $2$ $18$ $( 1,32)( 2,33)( 3,31)( 4,23)( 5,24)( 6,22)( 7,13)( 8,15)( 9,14)(10,28)(11,30)(12,29)(16,35)(17,36)(18,34)(19,27)(20,25)(21,26)$
3A $3^{12}$ $2$ $3$ $24$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,15,14)(16,17,18)(19,21,20)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,36,35)$
3B $3^{12}$ $2$ $3$ $24$ $( 1, 3, 2)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,26,27)(28,30,29)(31,33,32)(34,36,35)$
3C $3^{6},1^{18}$ $4$ $3$ $12$ $( 4, 6, 5)( 7, 9, 8)(16,17,18)(19,21,20)(28,29,30)(31,32,33)$
3D $3^{12}$ $6$ $3$ $24$ $( 1,26,15)( 2,25,13)( 3,27,14)( 4,30,18)( 5,29,17)( 6,28,16)( 7,21,31)( 8,19,33)( 9,20,32)(10,23,36)(11,24,35)(12,22,34)$
3E $3^{12}$ $6$ $3$ $24$ $( 1,14,25)( 2,15,27)( 3,13,26)( 4,16,29)( 5,18,28)( 6,17,30)( 7,32,19)( 8,31,20)( 9,33,21)(10,35,22)(11,34,23)(12,36,24)$
3F $3^{12}$ $6$ $3$ $24$ $( 1,27,13)( 2,26,14)( 3,25,15)( 4,28,17)( 5,30,16)( 6,29,18)( 7,20,33)( 8,21,32)( 9,19,31)(10,24,34)(11,22,36)(12,23,35)$
3G $3^{12}$ $6$ $3$ $24$ $( 1,26,14)( 2,25,15)( 3,27,13)( 4,17,29)( 5,16,28)( 6,18,30)( 7,20,32)( 8,21,31)( 9,19,33)(10,34,23)(11,36,24)(12,35,22)$
3H $3^{10},1^{6}$ $12$ $3$ $20$ $( 1, 3, 2)( 4,17,30)( 5,16,29)( 6,18,28)( 7, 9, 8)(10,36,24)(11,35,22)(12,34,23)(13,14,15)(19,20,21)$
3I $3^{12}$ $12$ $3$ $24$ $( 1,13,25)( 2,14,27)( 3,15,26)( 4,30,16)( 5,29,18)( 6,28,17)( 7,33,21)( 8,32,19)( 9,31,20)(10,24,35)(11,22,34)(12,23,36)$
3J $3^{10},1^{6}$ $12$ $3$ $20$ $( 1, 2, 3)( 4,17,30)( 5,16,29)( 6,18,28)( 7, 9, 8)(10,35,23)(11,34,24)(12,36,22)(19,20,21)(25,26,27)$
3K $3^{10},1^{6}$ $12$ $3$ $20$ $( 4,17,30)( 5,16,29)( 6,18,28)( 7, 9, 8)(10,34,22)(11,36,23)(12,35,24)(13,15,14)(19,20,21)(25,27,26)$
6A $6^{6}$ $18$ $6$ $30$ $( 1, 5,26,29,15,17)( 2, 6,25,28,13,16)( 3, 4,27,30,14,18)( 7,23,21,36,31,10)( 8,22,19,34,33,12)( 9,24,20,35,32,11)$
6B $6^{6}$ $18$ $6$ $30$ $( 1,18,14,28,25, 5)( 2,17,15,30,27, 6)( 3,16,13,29,26, 4)( 7,10,32,35,19,22)( 8,12,31,36,20,24)( 9,11,33,34,21,23)$
6C $6^{6}$ $18$ $6$ $30$ $( 1,29, 3,30, 2,28)( 4,13, 6,15, 5,14)( 7,35, 9,34, 8,36)(10,21,11,20,12,19)(16,26,17,27,18,25)(22,33,23,31,24,32)$
6D $6^{6}$ $18$ $6$ $30$ $( 1, 6,27,29,13,18)( 2, 4,26,28,14,17)( 3, 5,25,30,15,16)( 7,22,20,36,33,11)( 8,24,21,34,32,10)( 9,23,19,35,31,12)$
6E $6^{6}$ $54$ $6$ $30$ $( 1,36,26,24,14,11)( 2,35,25,22,15,12)( 3,34,27,23,13,10)( 4,33,17, 9,29,19)( 5,31,16, 8,28,21)( 6,32,18, 7,30,20)$
6F $6^{6}$ $54$ $6$ $30$ $( 1,33, 3,32, 2,31)( 4,22, 5,23, 6,24)( 7,14, 8,13, 9,15)(10,29,11,28,12,30)(16,36,18,35,17,34)(19,26,20,27,21,25)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 6A 6B 6C 6D 6E 6F
Size 1 9 27 27 2 2 4 6 6 6 6 12 12 12 12 18 18 18 18 54 54
2 P 1A 1A 1A 1A 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3D 3E 3A 3F 3G 3B
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2B 2C
Type
324.121.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
324.121.1b R 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1
324.121.1c R 1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 −1
324.121.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1
324.121.2a R 2 0 2 0 2 2 2 2 2 2 −1 −1 −1 −1 −1 0 0 0 0 −1 0
324.121.2b R 2 2 0 0 −1 2 −1 −1 −1 2 2 −1 −1 −1 2 −1 2 −1 −1 0 0
324.121.2c R 2 2 0 0 −1 2 −1 −1 2 −1 2 −1 −1 2 −1 −1 −1 2 −1 0 0
324.121.2d R 2 2 0 0 −1 2 −1 2 −1 −1 2 2 −1 −1 −1 2 −1 −1 −1 0 0
324.121.2e R 2 2 0 0 2 2 2 −1 −1 −1 2 −1 2 −1 −1 −1 −1 −1 2 0 0
324.121.2f R 2 −2 0 0 −1 2 −1 −1 −1 2 2 −1 −1 −1 2 1 −2 1 1 0 0
324.121.2g R 2 −2 0 0 −1 2 −1 −1 2 −1 2 −1 −1 2 −1 1 1 −2 1 0 0
324.121.2h R 2 −2 0 0 −1 2 −1 2 −1 −1 2 2 −1 −1 −1 −2 1 1 1 0 0
324.121.2i R 2 −2 0 0 2 2 2 −1 −1 −1 2 −1 2 −1 −1 1 1 1 −2 0 0
324.121.2j R 2 0 −2 0 2 2 2 2 2 2 −1 −1 −1 −1 −1 0 0 0 0 1 0
324.121.4a R 4 0 0 0 −2 4 −2 −2 −2 4 −2 1 1 1 −2 0 0 0 0 0 0
324.121.4b R 4 0 0 0 −2 4 −2 −2 4 −2 −2 1 1 −2 1 0 0 0 0 0 0
324.121.4c R 4 0 0 0 −2 4 −2 4 −2 −2 −2 −2 1 1 1 0 0 0 0 0 0
324.121.4d R 4 0 0 0 4 4 4 −2 −2 −2 −2 1 −2 1 1 0 0 0 0 0 0
324.121.6a R 6 0 0 2 6 −3 −3 0 0 0 0 0 0 0 0 0 0 0 0 0 −1
324.121.6b R 6 0 0 −2 6 −3 −3 0 0 0 0 0 0 0 0 0 0 0 0 0 1
324.121.12a R 12 0 0 0 −6 −6 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed