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Properties

Label 36T11
Degree $36$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2:C_9$

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Show commands: Magma

magma: G := TransitiveGroup(36, 11);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $11$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2:C_9$
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:  (1,8,11,4,5,10)(2,7,12,3,6,9)(13,19,22,14,20,21)(15,18,23,16,17,24)(25,32,36,28,30,34)(26,31,35,27,29,33), (1,27,13,5,31,20,11,33,22)(2,28,14,6,32,19,12,34,21)(3,26,15,7,29,17,9,35,23)(4,25,16,8,30,18,10,36,24)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$9$:  $C_9$
$12$:  $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: $A_4$

Degree 6: $A_4$

Degree 9: $C_9$

Degree 12: $A_4$

Degree 18: $C_2^2 : C_9$

Low degree siblings

18T7

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

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:  36.3
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 6A1 6A-1 9A1 9A-1 9A2 9A-2 9A4 9A-4
Size 1 3 1 1 3 3 4 4 4 4 4 4
2 P 1A 1A 3A-1 3A1 3A1 3A-1 9A-4 9A-2 9A4 9A-1 9A2 9A1
3 P 1A 2A 1A 1A 2A 2A 3A1 3A-1 3A-1 3A1 3A1 3A-1
Type
36.3.1a R 1 1 1 1 1 1 1 1 1 1 1 1
36.3.1b1 C 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
36.3.1b2 C 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
36.3.1c1 C 1 1 ζ9−3 ζ93 ζ93 ζ9−3 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2
36.3.1c2 C 1 1 ζ93 ζ9−3 ζ9−3 ζ93 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92
36.3.1c3 C 1 1 ζ9−3 ζ93 ζ93 ζ9−3 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9
36.3.1c4 C 1 1 ζ93 ζ9−3 ζ9−3 ζ93 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1
36.3.1c5 C 1 1 ζ9−3 ζ93 ζ93 ζ9−3 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94
36.3.1c6 C 1 1 ζ93 ζ9−3 ζ9−3 ζ93 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4
36.3.3a R 3 −1 3 3 −1 −1 0 0 0 0 0 0
36.3.3b1 C 3 −1 3ζ3−1 3ζ3 −ζ3 −ζ3−1 0 0 0 0 0 0
36.3.3b2 C 3 −1 3ζ3 3ζ3−1 −ζ3−1 −ζ3 0 0 0 0 0 0

magma: CharacterTable(G);