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Properties

Label 36T43
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3:D_{12}$

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Show commands: Magma

magma: G := TransitiveGroup(36, 43);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $43$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3:D_{12}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,6,35,3,7,34,2,5,36,4,8,33)(9,13,17,11,16,20,10,14,18,12,15,19)(21,27,31,23,26,29,22,28,32,24,25,30), (1,25,13)(2,26,14)(3,27,16)(4,28,15)(5,29,18)(6,30,17)(7,31,20)(8,32,19)(9,33,24)(10,34,23)(11,35,21)(12,36,22), (1,14)(2,13)(3,16)(4,15)(5,9)(6,10)(7,11)(8,12)(17,34)(18,33)(19,36)(20,35)(21,31)(22,32)(23,30)(24,29)(25,26)
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 4
$8$:  $D_{4}$
$12$:  $D_{6}$ x 4
$18$:  $C_3^2:C_2$
$24$:  $D_{12}$ x 4
$36$:  18T12

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 4

Degree 4: $D_{4}$

Degree 6: $D_{6}$ x 4

Degree 9: $C_3^2:C_2$

Degree 12: $D_{12}$ x 4

Degree 18: 18T12

Low degree siblings

36T43

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 $2^{18}$ $18$ $2$ $18$ $( 1,33)( 2,34)( 3,36)( 4,35)( 5, 7)( 6, 8)( 9,25)(10,26)(11,28)(12,27)(13,24)(14,23)(15,21)(16,22)(17,32)(18,31)(19,30)(20,29)$
2C $2^{17},1^{2}$ $18$ $2$ $17$ $( 3, 4)( 5,34)( 6,33)( 7,36)( 8,35)( 9,30)(10,29)(11,32)(12,31)(13,25)(14,26)(15,27)(16,28)(17,24)(18,23)(19,21)(20,22)$
3A $3^{12}$ $2$ $3$ $24$ $( 1,22,20)( 2,21,19)( 3,24,18)( 4,23,17)( 5,27, 9)( 6,28,10)( 7,25,12)( 8,26,11)(13,36,31)(14,35,32)(15,34,30)(16,33,29)$
3B $3^{12}$ $2$ $3$ $24$ $( 1,31,12)( 2,32,11)( 3,29, 9)( 4,30,10)( 5,24,16)( 6,23,15)( 7,22,13)( 8,21,14)(17,34,28)(18,33,27)(19,35,26)(20,36,25)$
3C $3^{12}$ $2$ $3$ $24$ $( 1, 7,36)( 2, 8,35)( 3, 5,33)( 4, 6,34)( 9,16,18)(10,15,17)(11,14,19)(12,13,20)(21,26,32)(22,25,31)(23,28,30)(24,27,29)$
3D $3^{12}$ $2$ $3$ $24$ $( 1,25,13)( 2,26,14)( 3,27,16)( 4,28,15)( 5,29,18)( 6,30,17)( 7,31,20)( 8,32,19)( 9,33,24)(10,34,23)(11,35,21)(12,36,22)$
4A $4^{9}$ $2$ $4$ $27$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,19,18,20)(21,24,22,23)(25,28,26,27)(29,31,30,32)(33,36,34,35)$
6A $6^{6}$ $2$ $6$ $30$ $( 1,32,12, 2,31,11)( 3,30, 9, 4,29,10)( 5,23,16, 6,24,15)( 7,21,13, 8,22,14)(17,33,28,18,34,27)(19,36,26,20,35,25)$
6B $6^{6}$ $2$ $6$ $30$ $( 1,35, 7, 2,36, 8)( 3,34, 5, 4,33, 6)( 9,17,16,10,18,15)(11,20,14,12,19,13)(21,31,26,22,32,25)(23,29,28,24,30,27)$
6C $6^{6}$ $2$ $6$ $30$ $( 1,21,20, 2,22,19)( 3,23,18, 4,24,17)( 5,28, 9, 6,27,10)( 7,26,12, 8,25,11)(13,35,31,14,36,32)(15,33,30,16,34,29)$
6D $6^{6}$ $2$ $6$ $30$ $( 1,26,13, 2,25,14)( 3,28,16, 4,27,15)( 5,30,18, 6,29,17)( 7,32,20, 8,31,19)( 9,34,24,10,33,23)(11,36,21,12,35,22)$
12A1 $12^{3}$ $2$ $12$ $33$ $( 1,30,11, 3,31,10, 2,29,12, 4,32, 9)( 5,22,15, 8,24,13, 6,21,16, 7,23,14)(17,35,27,20,34,26,18,36,28,19,33,25)$
12A5 $12^{3}$ $2$ $12$ $33$ $( 1,28,14, 3,25,15, 2,27,13, 4,26,16)( 5,31,17, 8,29,20, 6,32,18, 7,30,19)( 9,36,23,11,33,22,10,35,24,12,34,21)$
12B1 $12^{3}$ $2$ $12$ $33$ $( 1,29,11, 4,31, 9, 2,30,12, 3,32,10)( 5,21,15, 7,24,14, 6,22,16, 8,23,13)(17,36,27,19,34,25,18,35,28,20,33,26)$
12B5 $12^{3}$ $2$ $12$ $33$ $( 1,24,19, 4,22,18, 2,23,20, 3,21,17)( 5,26,10, 7,27,11, 6,25, 9, 8,28,12)(13,33,32,15,36,29,14,34,31,16,35,30)$
12C1 $12^{3}$ $2$ $12$ $33$ $( 1,23,19, 3,22,17, 2,24,20, 4,21,18)( 5,25,10, 8,27,12, 6,26, 9, 7,28,11)(13,34,32,16,36,30,14,33,31,15,35,29)$
12C5 $12^{3}$ $2$ $12$ $33$ $( 1,27,14, 4,25,16, 2,28,13, 3,26,15)( 5,32,17, 7,29,19, 6,31,18, 8,30,20)( 9,35,23,12,33,21,10,36,24,11,34,22)$
12D1 $12^{3}$ $2$ $12$ $33$ $( 1, 6,35, 3, 7,34, 2, 5,36, 4, 8,33)( 9,13,17,11,16,20,10,14,18,12,15,19)(21,27,31,23,26,29,22,28,32,24,25,30)$
12D5 $12^{3}$ $2$ $12$ $33$ $( 1,34, 8, 3,36, 6, 2,33, 7, 4,35, 5)( 9,20,15,11,18,13,10,19,16,12,17,14)(21,29,25,23,32,27,22,30,26,24,31,28)$

Malle's constant $a(G)$:     $1/17$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  72.33
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 3D 4A 6A 6B 6C 6D 12A1 12A5 12B1 12B5 12C1 12C5 12D1 12D5
Size 1 1 18 18 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3B 3A 3D 3C 2A 3A 3D 3B 3C 6A 6C 6A 6B 6B 6C 6D 6D
3 P 1A 2A 2B 2C 1A 1A 1A 1A 4A 2A 2A 2A 2A 4A 4A 4A 4A 4A 4A 4A 4A
Type
72.33.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.33.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.33.1c R 1 1 −1 1 1 1 1 1 −1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
72.33.1d R 1 1 1 −1 1 1 1 1 −1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
72.33.2a R 2 2 0 0 −1 −1 −1 2 2 −1 −1 −1 2 −1 −1 −1 −1 −1 −1 2 2
72.33.2b R 2 2 0 0 −1 −1 2 −1 2 −1 −1 2 −1 −1 −1 −1 −1 2 2 −1 −1
72.33.2c R 2 2 0 0 −1 2 −1 −1 2 −1 2 −1 −1 −1 −1 2 2 −1 −1 −1 −1
72.33.2d R 2 2 0 0 2 −1 −1 −1 2 2 −1 −1 −1 2 2 −1 −1 −1 −1 −1 −1
72.33.2e R 2 −2 0 0 2 2 2 2 0 −2 −2 −2 −2 0 0 0 0 0 0 0 0
72.33.2f R 2 2 0 0 −1 −1 −1 2 −2 −1 −1 −1 2 1 1 1 1 1 1 −2 −2
72.33.2g R 2 2 0 0 −1 −1 2 −1 −2 −1 −1 2 −1 1 1 1 1 −2 −2 1 1
72.33.2h R 2 2 0 0 −1 2 −1 −1 −2 −1 2 −1 −1 1 1 −2 −2 1 1 1 1
72.33.2i R 2 2 0 0 2 −1 −1 −1 −2 2 −1 −1 −1 −2 −2 1 1 1 1 1 1
72.33.2j1 R 2 −2 0 0 −1 −1 −1 2 0 1 1 1 −2 −ζ12−1−ζ12 ζ12−1+ζ12 −ζ12−1−ζ12 ζ12−1+ζ12 ζ12−1+ζ12 −ζ12−1−ζ12 0 0
72.33.2j2 R 2 −2 0 0 −1 −1 −1 2 0 1 1 1 −2 ζ12−1+ζ12 −ζ12−1−ζ12 ζ12−1+ζ12 −ζ12−1−ζ12 −ζ12−1−ζ12 ζ12−1+ζ12 0 0
72.33.2k1 R 2 −2 0 0 −1 −1 2 −1 0 1 1 −2 1 −ζ12−1−ζ12 ζ12−1+ζ12 ζ12−1+ζ12 −ζ12−1−ζ12 0 0 −ζ12−1−ζ12 ζ12−1+ζ12
72.33.2k2 R 2 −2 0 0 −1 −1 2 −1 0 1 1 −2 1 ζ12−1+ζ12 −ζ12−1−ζ12 −ζ12−1−ζ12 ζ12−1+ζ12 0 0 ζ12−1+ζ12 −ζ12−1−ζ12
72.33.2l1 R 2 −2 0 0 −1 2 −1 −1 0 1 −2 1 1 −ζ12−1−ζ12 ζ12−1+ζ12 0 0 −ζ12−1−ζ12 ζ12−1+ζ12 ζ12−1+ζ12 −ζ12−1−ζ12
72.33.2l2 R 2 −2 0 0 −1 2 −1 −1 0 1 −2 1 1 ζ12−1+ζ12 −ζ12−1−ζ12 0 0 ζ12−1+ζ12 −ζ12−1−ζ12 −ζ12−1−ζ12 ζ12−1+ζ12
72.33.2m1 R 2 −2 0 0 2 −1 −1 −1 0 −2 1 1 1 0 0 −ζ12−1−ζ12 ζ12−1+ζ12 −ζ12−1−ζ12 ζ12−1+ζ12 −ζ12−1−ζ12 ζ12−1+ζ12
72.33.2m2 R 2 −2 0 0 2 −1 −1 −1 0 −2 1 1 1 0 0 ζ12−1+ζ12 −ζ12−1−ζ12 ζ12−1+ζ12 −ζ12−1−ζ12 ζ12−1+ζ12 −ζ12−1−ζ12

magma: CharacterTable(G);