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Properties

Label 36T33
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, 33);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $33$
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:  (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,13)(8,14)(9,36)(10,35)(11,34)(12,33)(21,22)(25,31)(26,32)(27,29)(28,30), (1,23,7,25,33,31)(2,24,8,26,34,32)(3,21,5,27,36,30)(4,22,6,28,35,29)(9,19,16,11,18,14)(10,20,15,12,17,13)
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 2
$8$:  $D_{4}$
$12$:  $D_{6}$ x 2
$24$:  $D_{12}$, $(C_6\times C_2):C_2$
$36$:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 2

Degree 4: $D_{4}$

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

Degree 9: $S_3^2$

Degree 12: $D_{12}$, $(C_6\times C_2):C_2$, 12T38 x 2

Degree 18: $S_3^2$

Low degree siblings

12T38 x 2, 24T74, 36T38

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

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:  72.23
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 4A 6A 6B 6C 6D1 6D-1 12A1 12A5
Size 1 1 6 18 2 2 4 6 2 2 4 6 6 6 6
2 P 1A 1A 1A 1A 3A 3B 3C 2A 3B 3A 3C 3A 3A 6A 6A
3 P 1A 2A 2B 2C 1A 1A 1A 4A 2A 2A 2A 2B 2B 4A 4A
Type
72.23.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.23.1b R 1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 1 1
72.23.1c R 1 1 −1 1 1 1 1 −1 1 1 1 −1 −1 −1 −1
72.23.1d R 1 1 1 −1 1 1 1 −1 1 1 1 1 1 −1 −1
72.23.2a R 2 2 0 0 2 −1 −1 2 −1 2 −1 0 0 −1 −1
72.23.2b R 2 2 2 0 −1 2 −1 0 2 −1 −1 −1 −1 0 0
72.23.2c R 2 −2 0 0 2 2 2 0 −2 −2 −2 0 0 0 0
72.23.2d R 2 2 −2 0 −1 2 −1 0 2 −1 −1 1 1 0 0
72.23.2e R 2 2 0 0 2 −1 −1 −2 −1 2 −1 0 0 1 1
72.23.2f1 R 2 −2 0 0 2 −1 −1 0 1 −2 1 0 0 −ζ12−1−ζ12 ζ12−1+ζ12
72.23.2f2 R 2 −2 0 0 2 −1 −1 0 1 −2 1 0 0 ζ12−1+ζ12 −ζ12−1−ζ12
72.23.2g1 C 2 −2 0 0 −1 2 −1 0 −2 1 1 −1−2ζ3 1+2ζ3 0 0
72.23.2g2 C 2 −2 0 0 −1 2 −1 0 −2 1 1 1+2ζ3 −1−2ζ3 0 0
72.23.4a R 4 4 0 0 −2 −2 1 0 −2 −2 1 0 0 0 0
72.23.4b R 4 −4 0 0 −2 −2 1 0 2 2 −1 0 0 0 0

magma: CharacterTable(G);