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Properties

Label 36T9
Degree $36$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_9:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(36, 9);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_9:C_4$
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:  (1,3,2,4)(5,34,6,33)(7,35,8,36)(9,32,10,31)(11,30,12,29)(13,28,14,27)(15,26,16,25)(17,23,18,24)(19,21,20,22), (1,34,32,28,24,19,16,12,7)(2,33,31,27,23,20,15,11,8)(3,35,29,25,21,17,14,10,6)(4,36,30,26,22,18,13,9,5)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$6$:  $S_3$
$12$:  $C_3 : C_4$
$18$:  $D_{9}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $S_3$

Degree 9: $D_{9}$

Degree 12: $C_3 : C_4$

Degree 18: $D_9$

Low degree siblings

There are no siblings 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 $2^{18}$ $1$ $2$ $18$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
3A $3^{12}$ $2$ $3$ $24$ $( 1,28,16)( 2,27,15)( 3,25,14)( 4,26,13)( 5,30,18)( 6,29,17)( 7,32,19)( 8,31,20)( 9,36,22)(10,35,21)(11,33,23)(12,34,24)$
4A1 $4^{9}$ $9$ $4$ $27$ $( 1, 3, 2, 4)( 5,34, 6,33)( 7,35, 8,36)( 9,32,10,31)(11,30,12,29)(13,28,14,27)(15,26,16,25)(17,23,18,24)(19,21,20,22)$
4A-1 $4^{9}$ $9$ $4$ $27$ $( 1, 4, 2, 3)( 5,33, 6,34)( 7,36, 8,35)( 9,31,10,32)(11,29,12,30)(13,27,14,28)(15,25,16,26)(17,24,18,23)(19,22,20,21)$
6A $6^{6}$ $2$ $6$ $30$ $( 1,27,16, 2,28,15)( 3,26,14, 4,25,13)( 5,29,18, 6,30,17)( 7,31,19, 8,32,20)( 9,35,22,10,36,21)(11,34,23,12,33,24)$
9A1 $9^{4}$ $2$ $9$ $32$ $( 1,34,32,28,24,19,16,12, 7)( 2,33,31,27,23,20,15,11, 8)( 3,35,29,25,21,17,14,10, 6)( 4,36,30,26,22,18,13, 9, 5)$
9A2 $9^{4}$ $2$ $9$ $32$ $( 1,12,19,28,34, 7,16,24,32)( 2,11,20,27,33, 8,15,23,31)( 3,10,17,25,35, 6,14,21,29)( 4, 9,18,26,36, 5,13,22,30)$
9A4 $9^{4}$ $2$ $9$ $32$ $( 1,24, 7,28,12,32,16,34,19)( 2,23, 8,27,11,31,15,33,20)( 3,21, 6,25,10,29,14,35,17)( 4,22, 5,26, 9,30,13,36,18)$
18A1 $18^{2}$ $2$ $18$ $34$ $( 1,11,19,27,34, 8,16,23,32, 2,12,20,28,33, 7,15,24,31)( 3, 9,17,26,35, 5,14,22,29, 4,10,18,25,36, 6,13,21,30)$
18A5 $18^{2}$ $2$ $18$ $34$ $( 1,33,32,27,24,20,16,11, 7, 2,34,31,28,23,19,15,12, 8)( 3,36,29,26,21,18,14, 9, 6, 4,35,30,25,22,17,13,10, 5)$
18A7 $18^{2}$ $2$ $18$ $34$ $( 1,23, 7,27,12,31,16,33,19, 2,24, 8,28,11,32,15,34,20)( 3,22, 6,26,10,30,14,36,17, 4,21, 5,25, 9,29,13,35,18)$

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:  36.1
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 4A1 4A-1 6A 9A1 9A2 9A4 18A1 18A5 18A7
Size 1 1 2 9 9 2 2 2 2 2 2 2
2 P 1A 1A 3A 2A 2A 3A 9A2 9A4 9A1 9A4 9A2 9A1
3 P 1A 2A 1A 4A-1 4A1 2A 3A 3A 3A 6A 6A 6A
Type
36.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1
36.1.1b R 1 1 1 −1 −1 1 1 1 1 1 1 1
36.1.1c1 C 1 −1 1 −i i −1 1 1 1 −1 −1 −1
36.1.1c2 C 1 −1 1 i −i −1 1 1 1 −1 −1 −1
36.1.2a R 2 2 2 0 0 2 −1 −1 −1 −1 −1 −1
36.1.2b S 2 −2 2 0 0 −2 −1 −1 −1 1 1 1
36.1.2c1 R 2 2 −1 0 0 −1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94
36.1.2c2 R 2 2 −1 0 0 −1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92
36.1.2c3 R 2 2 −1 0 0 −1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9
36.1.2d1 S 2 −2 −1 0 0 1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94
36.1.2d2 S 2 −2 −1 0 0 1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92
36.1.2d3 S 2 −2 −1 0 0 1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9

magma: CharacterTable(G);