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Properties

Label 36T47
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{36}$

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Show commands: Magma

magma: G := TransitiveGroup(36, 47);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $47$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{36}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,7)(2,8)(3,6)(4,5)(9,36)(10,35)(11,33)(12,34)(13,31)(14,32)(15,30)(16,29)(17,25)(18,26)(19,27)(20,28)(21,23)(22,24), (1,36,31,28,24,18,16,12,5,4,34,29,26,22,20,13,9,7,2,35,32,27,23,17,15,11,6,3,33,30,25,21,19,14,10,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$
$12$:  $D_{6}$
$18$:  $D_{9}$
$24$:  $D_{12}$
$36$:  $D_{18}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $D_{6}$

Degree 9: $D_{9}$

Degree 12: $D_{12}$

Degree 18: $D_{18}$

Low degree siblings

36T47

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  72.6
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);