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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

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

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);