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Properties

Label 27T3
Degree $27$
Order $27$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $\He_3$

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Show commands: Magma

magma: G := TransitiveGroup(27, 3);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $3$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\He_3$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $27$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,21,10)(2,19,11)(3,20,12)(4,24,13)(5,22,14)(6,23,15)(7,25,18)(8,26,16)(9,27,17), (1,25,4)(2,26,5)(3,27,6)(7,8,9)(10,11,12)(13,14,15)(16,21,23)(17,19,24)(18,20,22)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$ x 4

Degree 9: $C_3^2$, $C_3^2:C_3$ x 4

Low degree siblings

9T7 x 4

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $27=3^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $2$
Label:  27.3
magma: IdentifyGroup(G);
 
Character table:

1A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 3E1 3E-1
Size 1 1 1 3 3 3 3 3 3 3 3
3 P 1A 3A-1 3A1 3B-1 3D-1 3C-1 3D1 3C1 3E-1 3B1 3E1
Type
27.3.1a R 1 1 1 1 1 1 1 1 1 1 1
27.3.1b1 C 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1
27.3.1b2 C 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1
27.3.1c1 C 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3
27.3.1c2 C 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1
27.3.1d1 C 1 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1
27.3.1d2 C 1 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3
27.3.1e1 C 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
27.3.1e2 C 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
27.3.3a1 C 3 3ζ3−1 3ζ3 0 0 0 0 0 0 0 0
27.3.3a2 C 3 3ζ3 3ζ3−1 0 0 0 0 0 0 0 0

magma: CharacterTable(G);