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Properties

Label 36T14
Degree $36$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(36, 14);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $14$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^2:C_4$
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:  (1,16,6,30)(2,15,5,29)(3,13,8,32)(4,14,7,31)(9,27,21,19)(10,28,22,20)(11,26,24,18)(12,25,23,17)(33,35,34,36), (1,8,28,21)(2,7,27,22)(3,5,26,23)(4,6,25,24)(9,31,18,33)(10,32,17,34)(11,29,20,36)(12,30,19,35)(13,16,14,15)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: $C_3^2:C_4$ x 2

Degree 9: $C_3^2:C_4$

Degree 12: $(C_3\times C_3):C_4$ x 2

Degree 18: $C_3^2 : C_4$

Low degree siblings

6T10 x 2, 9T9, 12T17 x 2, 18T10

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

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:  36.9
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 3B 4A1 4A-1
Size 1 9 4 4 9 9
2 P 1A 1A 3A 3B 2A 2A
3 P 1A 2A 1A 1A 4A-1 4A1
Type
36.9.1a R 1 1 1 1 1 1
36.9.1b R 1 1 1 1 −1 −1
36.9.1c1 C 1 −1 1 1 −i i
36.9.1c2 C 1 −1 1 1 i −i
36.9.4a R 4 0 −2 1 0 0
36.9.4b R 4 0 1 −2 0 0

magma: CharacterTable(G);