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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

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

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);