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Properties

Label 27T16
Degree $27$
Order $81$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_9:C_3^2$

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Show commands: Magma

magma: G := TransitiveGroup(27, 16);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $16$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_9:C_3^2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $9$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,9,13,20,27,4,11,17,23)(2,7,14,21,25,5,12,18,24)(3,8,15,19,26,6,10,16,22), (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), (4,23,13)(5,24,14)(6,22,15)(7,18,25)(8,16,26)(9,17,27)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$ x 13
$9$:  $C_3^2$ x 13
$27$:  $C_9:C_3$ x 3, 27T4

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$ x 4

Degree 9: $C_3^2$, $C_9:C_3$ x 3

Low degree siblings

27T16 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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $3$ $( 4,13,23)( 5,14,24)( 6,15,22)( 7,25,18)( 8,26,16)( 9,27,17)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $3$ $( 4,23,13)( 5,24,14)( 6,22,15)( 7,18,25)( 8,16,26)( 9,17,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1, 2, 3)( 4,14,22)( 5,15,23)( 6,13,24)( 7,26,17)( 8,27,18)( 9,25,16) (10,11,12)(19,20,21)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1, 2, 3)( 4,24,15)( 5,22,13)( 6,23,14)( 7,16,27)( 8,17,25)( 9,18,26) (10,11,12)(19,20,21)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1, 3, 2)( 4,15,24)( 5,13,22)( 6,14,23)( 7,27,16)( 8,25,17)( 9,26,18) (10,12,11)(19,21,20)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $3$ $3$ $( 1, 3, 2)( 4,22,14)( 5,23,15)( 6,24,13)( 7,17,26)( 8,18,27)( 9,16,25) (10,12,11)(19,21,20)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 4, 9,11,13,17,20,23,27)( 2, 5, 7,12,14,18,21,24,25)( 3, 6, 8,10,15,16,19, 22,26)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 4,17,11,13,27,20,23, 9)( 2, 5,18,12,14,25,21,24, 7)( 3, 6,16,10,15,26,19, 22, 8)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 4,27,11,13, 9,20,23,17)( 2, 5,25,12,14, 7,21,24,18)( 3, 6,26,10,15, 8,19, 22,16)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 5, 8,11,14,16,20,24,26)( 2, 6, 9,12,15,17,21,22,27)( 3, 4, 7,10,13,18,19, 23,25)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 5,16,11,14,26,20,24, 8)( 2, 6,17,12,15,27,21,22, 9)( 3, 4,18,10,13,25,19, 23, 7)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 5,26,11,14, 8,20,24,16)( 2, 6,27,12,15, 9,21,22,17)( 3, 4,25,10,13, 7,19, 23,18)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 6, 7,11,15,18,20,22,25)( 2, 4, 8,12,13,16,21,23,26)( 3, 5, 9,10,14,17,19, 24,27)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 6,18,11,15,25,20,22, 7)( 2, 4,16,12,13,26,21,23, 8)( 3, 5,17,10,14,27,19, 24, 9)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 6,25,11,15, 7,20,22,18)( 2, 4,26,12,13, 8,21,23,16)( 3, 5,27,10,14, 9,19, 24,17)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 7,22,20,25,15,11,18, 6)( 2, 8,23,21,26,13,12,16, 4)( 3, 9,24,19,27,14,10, 17, 5)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 7,15,20,25, 6,11,18,22)( 2, 8,13,21,26, 4,12,16,23)( 3, 9,14,19,27, 5,10, 17,24)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 7, 6,20,25,22,11,18,15)( 2, 8, 4,21,26,23,12,16,13)( 3, 9, 5,19,27,24,10, 17,14)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 8,24,20,26,14,11,16, 5)( 2, 9,22,21,27,15,12,17, 6)( 3, 7,23,19,25,13,10, 18, 4)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 8,14,20,26, 5,11,16,24)( 2, 9,15,21,27, 6,12,17,22)( 3, 7,13,19,25, 4,10, 18,23)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 8, 5,20,26,24,11,16,14)( 2, 9, 6,21,27,22,12,17,15)( 3, 7, 4,19,25,23,10, 18,13)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 9,23,20,27,13,11,17, 4)( 2, 7,24,21,25,14,12,18, 5)( 3, 8,22,19,26,15,10, 16, 6)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 9,13,20,27, 4,11,17,23)( 2, 7,14,21,25, 5,12,18,24)( 3, 8,15,19,26, 6,10, 16,22)$
$ 9, 9, 9 $ $3$ $9$ $( 1, 9, 4,20,27,23,11,17,13)( 2, 7, 5,21,25,24,12,18,14)( 3, 8, 6,19,26,22,10, 16,15)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,10,21)( 2,11,19)( 3,12,20)( 4,15,24)( 5,13,22)( 6,14,23)( 7,17,26) ( 8,18,27)( 9,16,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,11,20)( 2,12,21)( 3,10,19)( 4,13,23)( 5,14,24)( 6,15,22)( 7,18,25) ( 8,16,26)( 9,17,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,12,19)( 2,10,20)( 3,11,21)( 4,14,22)( 5,15,23)( 6,13,24)( 7,16,27) ( 8,17,25)( 9,18,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,19,12)( 2,20,10)( 3,21,11)( 4,22,14)( 5,23,15)( 6,24,13)( 7,27,16) ( 8,25,17)( 9,26,18)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,20,11)( 2,21,12)( 3,19,10)( 4,23,13)( 5,24,14)( 6,22,15)( 7,25,18) ( 8,26,16)( 9,27,17)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,21,10)( 2,19,11)( 3,20,12)( 4,24,15)( 5,22,13)( 6,23,14)( 7,26,17) ( 8,27,18)( 9,25,16)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $81=3^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $2$
Label:  81.13
magma: IdentifyGroup(G);
 
Character table:    33 x 33 character table

magma: CharacterTable(G);