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Properties

Label 36T8
Degree $36$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6:S_3$

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Show commands: Magma

magma: G := TransitiveGroup(36, 8);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $8$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_6:S_3$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $36$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,4)(2,3)(5,34)(6,33)(7,35)(8,36)(9,31)(10,32)(11,30)(12,29)(13,28)(14,27)(15,25)(16,26)(17,23)(18,24)(19,21)(20,22), (1,11)(2,12)(3,10)(4,9)(5,18)(6,17)(7,20)(8,19)(13,34)(14,33)(15,35)(16,36)(21,25)(22,26)(23,28)(24,27)(29,31)(30,32), (1,34,7)(2,33,8)(3,36,6)(4,35,5)(9,18,15)(10,17,16)(11,20,13)(12,19,14)(21,29,27)(22,30,28)(23,32,26)(24,31,25)
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
$12$:  $D_{6}$ x 4
$18$:  $C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$ x 4

Degree 4: $C_2^2$

Degree 6: $S_3$ x 4, $D_{6}$ x 8

Degree 9: $C_3^2:C_2$

Degree 12: $D_6$ x 4

Degree 18: $C_3^2 : C_2$, 18T12 x 2

Low degree siblings

18T12 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 $2^{18}$ $1$ $2$ $18$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
2B $2^{18}$ $9$ $2$ $18$ $( 1,36)( 2,35)( 3,34)( 4,33)( 5, 8)( 6, 7)( 9,26)(10,25)(11,27)(12,28)(13,21)(14,22)(15,23)(16,24)(17,31)(18,32)(19,30)(20,29)$
2C $2^{18}$ $9$ $2$ $18$ $( 1, 4)( 2, 3)( 5,34)( 6,33)( 7,35)( 8,36)( 9,31)(10,32)(11,30)(12,29)(13,28)(14,27)(15,25)(16,26)(17,23)(18,24)(19,21)(20,22)$
3A $3^{12}$ $2$ $3$ $24$ $( 1,16,27)( 2,15,28)( 3,13,25)( 4,14,26)( 5,19,32)( 6,20,31)( 7,17,29)( 8,18,30)( 9,22,33)(10,21,34)(11,24,36)(12,23,35)$
3B $3^{12}$ $2$ $3$ $24$ $( 1,34, 7)( 2,33, 8)( 3,36, 6)( 4,35, 5)( 9,18,15)(10,17,16)(11,20,13)(12,19,14)(21,29,27)(22,30,28)(23,32,26)(24,31,25)$
3C $3^{12}$ $2$ $3$ $24$ $( 1,17,21)( 2,18,22)( 3,20,24)( 4,19,23)( 5,12,26)( 6,11,25)( 7,10,27)( 8, 9,28)(13,31,36)(14,32,35)(15,30,33)(16,29,34)$
3D $3^{12}$ $2$ $3$ $24$ $( 1,10,29)( 2, 9,30)( 3,11,31)( 4,12,32)( 5,14,23)( 6,13,24)( 7,16,21)( 8,15,22)(17,27,34)(18,28,33)(19,26,35)(20,25,36)$
6A $6^{6}$ $2$ $6$ $30$ $( 1, 8,34, 2, 7,33)( 3, 5,36, 4, 6,35)( 9,16,18,10,15,17)(11,14,20,12,13,19)(21,28,29,22,27,30)(23,25,32,24,26,31)$
6B $6^{6}$ $2$ $6$ $30$ $( 1,15,27, 2,16,28)( 3,14,25, 4,13,26)( 5,20,32, 6,19,31)( 7,18,29, 8,17,30)( 9,21,33,10,22,34)(11,23,36,12,24,35)$
6C $6^{6}$ $2$ $6$ $30$ $( 1, 9,29, 2,10,30)( 3,12,31, 4,11,32)( 5,13,23, 6,14,24)( 7,15,21, 8,16,22)(17,28,34,18,27,33)(19,25,35,20,26,36)$
6D $6^{6}$ $2$ $6$ $30$ $( 1,18,21, 2,17,22)( 3,19,24, 4,20,23)( 5,11,26, 6,12,25)( 7, 9,27, 8,10,28)(13,32,36,14,31,35)(15,29,33,16,30,34)$

Malle's constant $a(G)$:     $1/18$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  36.13
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 3D 6A 6B 6C 6D
Size 1 1 9 9 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3B 3A 3D 3C 3A 3B 3C 3D
3 P 1A 2A 2B 2C 1A 1A 1A 1A 2A 2A 2A 2A
Type
36.13.1a R 1 1 1 1 1 1 1 1 1 1 1 1
36.13.1b R 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1
36.13.1c R 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1
36.13.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1
36.13.2a R 2 2 0 0 −1 −1 −1 2 −1 −1 −1 2
36.13.2b R 2 2 0 0 −1 −1 2 −1 −1 −1 2 −1
36.13.2c R 2 2 0 0 −1 2 −1 −1 −1 2 −1 −1
36.13.2d R 2 2 0 0 2 −1 −1 −1 2 −1 −1 −1
36.13.2e R 2 −2 0 0 −1 −1 −1 2 1 1 1 −2
36.13.2f R 2 −2 0 0 −1 −1 2 −1 1 1 −2 1
36.13.2g R 2 −2 0 0 −1 2 −1 −1 1 −2 1 1
36.13.2h R 2 −2 0 0 2 −1 −1 −1 −2 1 1 1

magma: CharacterTable(G);