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Properties

Label 36T73
36T73 1 3 1->3 15 1->15 2 4 2->4 16 2->16 13 3->13 14 4->14 5 19 5->19 34 5->34 6 20 6->20 33 6->33 7 17 7->17 35 7->35 8 18 8->18 36 8->36 9 24 9->24 27 9->27 10 23 10->23 28 10->28 11 22 11->22 25 11->25 12 21 12->21 26 12->26 30 13->30 32 13->32 29 14->29 31 14->31 15->29 15->31 16->30 16->32 17->22 17->25 18->21 18->26 19->24 19->27 20->23 20->28 21->7 21->13 22->8 22->14 23->5 23->16 24->6 24->15 25->9 25->35 26->10 26->36 27->11 27->34 28->12 28->33 29->2 29->19 30->1 30->20 31->3 31->17 32->4 32->18 33->9 34->10 35->12 36->11
Degree $36$
Order $108$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2:D_6$

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Copy content magma:G := TransitiveGroup(36, 73);
 

Group invariants

Abstract group:  $C_3^2:D_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $108=2^{2} \cdot 3^{3}$
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 $t$:  $73$
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:  $(1,3)(2,4)(5,34)(6,33)(7,35)(8,36)(9,24,15,31,17,25)(10,23,16,32,18,26)(11,22,14,29,19,27)(12,21,13,30,20,28)$, $(1,15,29,2,16,30)(3,13,32,4,14,31)(5,19,24,6,20,23)(7,17,22,8,18,21)(9,27,34,10,28,33)(11,25,35,12,26,36)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$36$:  $C_6\times S_3$
$54$:  $C_3^2 : C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $S_3$, $D_{6}$ x 2

Degree 9: $C_3^2 : C_6$

Degree 12: $D_6$

Degree 18: 18T21, 18T41 x 2

Low degree siblings

18T41 x 2, 18T42 x 2, 36T71, 36T75

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 $2^{18}$ $9$ $2$ $18$ $( 1, 3)( 2, 4)( 5,34)( 6,33)( 7,35)( 8,36)( 9,31)(10,32)(11,29)(12,30)(13,28)(14,27)(15,25)(16,26)(17,24)(18,23)(19,22)(20,21)$
2C $2^{18}$ $9$ $2$ $18$ $( 1,36)( 2,35)( 3,34)( 4,33)( 5, 7)( 6, 8)( 9,26)(10,25)(11,28)(12,27)(13,22)(14,21)(15,23)(16,24)(17,32)(18,31)(19,30)(20,29)$
3A $3^{12}$ $2$ $3$ $24$ $( 1, 7,33)( 2, 8,34)( 3, 6,35)( 4, 5,36)( 9,15,17)(10,16,18)(11,14,19)(12,13,20)(21,28,30)(22,27,29)(23,26,32)(24,25,31)$
3B1 $3^{8},1^{12}$ $3$ $3$ $16$ $( 1, 7,33)( 2, 8,34)( 3, 6,35)( 4, 5,36)(21,30,28)(22,29,27)(23,32,26)(24,31,25)$
3B-1 $3^{8},1^{12}$ $3$ $3$ $16$ $( 1,33, 7)( 2,34, 8)( 3,35, 6)( 4,36, 5)(21,28,30)(22,27,29)(23,26,32)(24,25,31)$
3C $3^{12}$ $6$ $3$ $24$ $( 1,29,10)( 2,30, 9)( 3,32,11)( 4,31,12)( 5,24,13)( 6,23,14)( 7,22,16)( 8,21,15)(17,34,28)(18,33,27)(19,35,26)(20,36,25)$
3D1 $3^{12}$ $6$ $3$ $24$ $( 1,22,10)( 2,21, 9)( 3,23,11)( 4,24,12)( 5,25,13)( 6,26,14)( 7,27,16)( 8,28,15)(17,34,30)(18,33,29)(19,35,32)(20,36,31)$
3D-1 $3^{12}$ $6$ $3$ $24$ $( 1,10,22)( 2, 9,21)( 3,11,23)( 4,12,24)( 5,13,25)( 6,14,26)( 7,16,27)( 8,15,28)(17,30,34)(18,29,33)(19,32,35)(20,31,36)$
6A $6^{6}$ $2$ $6$ $30$ $( 1,34, 7, 2,33, 8)( 3,36, 6, 4,35, 5)( 9,18,15,10,17,16)(11,20,14,12,19,13)(21,29,28,22,30,27)(23,31,26,24,32,25)$
6B1 $6^{4},2^{6}$ $3$ $6$ $26$ $( 1,34, 7, 2,33, 8)( 3,36, 6, 4,35, 5)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,27,30,22,28,29)(23,25,32,24,26,31)$
6B-1 $6^{4},2^{6}$ $3$ $6$ $26$ $( 1,34, 7, 2,33, 8)( 3,36, 6, 4,35, 5)( 9,16,17,10,15,18)(11,13,19,12,14,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
6C $6^{6}$ $6$ $6$ $30$ $( 1, 9,29, 2,10,30)( 3,12,32, 4,11,31)( 5,14,24, 6,13,23)( 7,15,22, 8,16,21)(17,27,34,18,28,33)(19,25,35,20,26,36)$
6D1 $6^{6}$ $6$ $6$ $30$ $( 1, 9,22, 2,10,21)( 3,12,23, 4,11,24)( 5,14,25, 6,13,26)( 7,15,27, 8,16,28)(17,29,34,18,30,33)(19,31,35,20,32,36)$
6D-1 $6^{6}$ $6$ $6$ $30$ $( 1, 9,27, 2,10,28)( 3,12,26, 4,11,25)( 5,14,31, 6,13,32)( 7,15,29, 8,16,30)(17,22,34,18,21,33)(19,24,35,20,23,36)$
6E1 $6^{4},2^{6}$ $9$ $6$ $26$ $( 1, 3)( 2, 4)( 5,34)( 6,33)( 7,35)( 8,36)( 9,24,15,31,17,25)(10,23,16,32,18,26)(11,22,14,29,19,27)(12,21,13,30,20,28)$
6E-1 $6^{4},2^{6}$ $9$ $6$ $26$ $( 1, 3)( 2, 4)( 5,34)( 6,33)( 7,35)( 8,36)( 9,25,17,31,15,24)(10,26,18,32,16,23)(11,27,19,29,14,22)(12,28,20,30,13,21)$
6F1 $6^{4},2^{6}$ $9$ $6$ $26$ $( 1,36)( 2,35)( 3,34)( 4,33)( 5, 7)( 6, 8)( 9,32,15,26,17,23)(10,31,16,25,18,24)(11,30,14,28,19,21)(12,29,13,27,20,22)$
6F-1 $6^{4},2^{6}$ $9$ $6$ $26$ $( 1,36)( 2,35)( 3,34)( 4,33)( 5, 7)( 6, 8)( 9,23,17,26,15,32)(10,24,18,25,16,31)(11,21,19,28,14,30)(12,22,20,27,13,29)$

Malle's constant $a(G)$:     $1/16$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 3B1 3B-1 3C 3D1 3D-1 6A 6B1 6B-1 6C 6D1 6D-1 6E1 6E-1 6F1 6F-1
Size 1 1 9 9 2 3 3 6 6 6 2 3 3 6 6 6 9 9 9 9
2 P 1A 1A 1A 1A 3A 3B-1 3B1 3C 3D-1 3D1 3A 3B1 3B-1 3C 3D1 3D-1 3B-1 3B1 3B-1 3B1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2B 2B 2C 2C
Type
108.25.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
108.25.1b R 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1
108.25.1c R 1 −1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 −1 −1
108.25.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
108.25.1e1 C 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3−1 ζ3
108.25.1e2 C 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3 ζ3−1
108.25.1f1 C 1 −1 −1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3 ζ3−1 ζ3
108.25.1f2 C 1 −1 −1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1 ζ3 ζ3−1
108.25.1g1 C 1 −1 1 −1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3
108.25.1g2 C 1 −1 1 −1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −1 ζ3 ζ3−1 −ζ3 −ζ3−1
108.25.1h1 C 1 1 −1 −1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 −ζ3−1 −ζ3 −ζ3−1 −ζ3
108.25.1h2 C 1 1 −1 −1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 −ζ3 −ζ3−1 −ζ3 −ζ3−1
108.25.2a R 2 2 0 0 2 2 2 −1 −1 −1 2 2 2 −1 −1 −1 0 0 0 0
108.25.2b R 2 −2 0 0 2 2 2 −1 −1 −1 −2 −2 −2 1 1 1 0 0 0 0
108.25.2c1 C 2 2 0 0 2 2ζ3−1 2ζ3 −ζ3−1 −ζ3 −1 2 2ζ3 2ζ3−1 −ζ3 −ζ3−1 −1 0 0 0 0
108.25.2c2 C 2 2 0 0 2 2ζ3 2ζ3−1 −ζ3 −ζ3−1 −1 2 2ζ3−1 2ζ3 −ζ3−1 −ζ3 −1 0 0 0 0
108.25.2d1 C 2 −2 0 0 2 2ζ3−1 2ζ3 −ζ3−1 −ζ3 −1 −2 −2ζ3 −2ζ3−1 ζ3 ζ3−1 1 0 0 0 0
108.25.2d2 C 2 −2 0 0 2 2ζ3 2ζ3−1 −ζ3 −ζ3−1 −1 −2 −2ζ3−1 −2ζ3 ζ3−1 ζ3 1 0 0 0 0
108.25.6a R 6 6 0 0 −3 0 0 0 0 0 −3 0 0 0 0 0 0 0 0 0
108.25.6b R 6 −6 0 0 −3 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed