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Properties

Label 36T10
Degree $36$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{18}$

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Show commands: Magma

magma: G := TransitiveGroup(36, 10);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$18$:  $D_{9}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $S_3$, $D_{6}$ x 2

Degree 9: $D_{9}$

Degree 12: $D_6$

Degree 18: $D_9$, $D_{18}$ x 2

Low degree siblings

18T13 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 $2^{18}$ $9$ $2$ $18$ $( 1,30)( 2,29)( 3,31)( 4,32)( 5,28)( 6,27)( 7,26)( 8,25)( 9,24)(10,23)(11,21)(12,22)(13,19)(14,20)(15,17)(16,18)(33,35)(34,36)$
2C $2^{18}$ $9$ $2$ $18$ $( 1,35)( 2,36)( 3,34)( 4,33)( 5,31)( 6,32)( 7,29)( 8,30)( 9,27)(10,28)(11,26)(12,25)(13,23)(14,24)(15,22)(16,21)(17,19)(18,20)$
3A $3^{12}$ $2$ $3$ $24$ $( 1,16,28)( 2,15,27)( 3,14,25)( 4,13,26)( 5,18,30)( 6,17,29)( 7,19,32)( 8,20,31)( 9,22,36)(10,21,35)(11,23,33)(12,24,34)$
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 $9^{4}$ $2$ $9$ $32$ $( 1,32,24,16, 7,34,28,19,12)( 2,31,23,15, 8,33,27,20,11)( 3,29,21,14, 6,35,25,17,10)( 4,30,22,13, 5,36,26,18, 9)$
9A2 $9^{4}$ $2$ $9$ $32$ $( 1,19,34,16,32,12,28, 7,24)( 2,20,33,15,31,11,27, 8,23)( 3,17,35,14,29,10,25, 6,21)( 4,18,36,13,30, 9,26, 5,22)$
9A4 $9^{4}$ $2$ $9$ $32$ $( 1, 7,12,16,19,24,28,32,34)( 2, 8,11,15,20,23,27,31,33)( 3, 6,10,14,17,21,25,29,35)( 4, 5, 9,13,18,22,26,30,36)$
18A1 $18^{2}$ $2$ $18$ $34$ $( 1,33,32,27,24,20,16,11, 7, 2,34,31,28,23,19,15,12, 8)( 3,36,29,26,21,18,14, 9, 6, 4,35,30,25,22,17,13,10, 5)$
18A5 $18^{2}$ $2$ $18$ $34$ $( 1,23, 7,27,12,31,16,33,19, 2,24, 8,28,11,32,15,34,20)( 3,22, 6,26,10,30,14,36,17, 4,21, 5,25, 9,29,13,35,18)$
18A7 $18^{2}$ $2$ $18$ $34$ $( 1,11,19,27,34, 8,16,23,32, 2,12,20,28,33, 7,15,24,31)( 3, 9,17,26,35, 5,14,22,29, 4,10,18,25,36, 6,13,21,30)$

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:  36.4
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 6A 9A1 9A2 9A4 18A1 18A5 18A7
Size 1 1 9 9 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3A 3A 9A2 9A4 9A1 9A1 9A4 9A2
3 P 1A 2A 2B 2C 1A 2A 3A 3A 3A 6A 6A 6A
Type
36.4.1a R 1 1 1 1 1 1 1 1 1 1 1 1
36.4.1b R 1 −1 −1 1 1 −1 1 1 1 −1 −1 −1
36.4.1c R 1 −1 1 −1 1 −1 1 1 1 −1 −1 −1
36.4.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1
36.4.2a R 2 2 0 0 2 2 −1 −1 −1 −1 −1 −1
36.4.2b R 2 −2 0 0 2 −2 −1 −1 −1 1 1 1
36.4.2c1 R 2 2 0 0 −1 −1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94
36.4.2c2 R 2 2 0 0 −1 −1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92
36.4.2c3 R 2 2 0 0 −1 −1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9
36.4.2d1 R 2 −2 0 0 −1 1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94
36.4.2d2 R 2 −2 0 0 −1 1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92
36.4.2d3 R 2 −2 0 0 −1 1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9

magma: CharacterTable(G);