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Properties

Label 27T19
Degree $27$
Order $81$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_3\wr C_3$

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Show commands: Magma

magma: G := TransitiveGroup(27, 19);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $19$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\wr C_3$
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:  (1,4,25)(2,5,26)(3,6,27)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,10,21)(2,11,19)(3,12,20)(4,13,24)(5,14,22)(6,15,23)(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 4
$9$:  $C_3^2$
$27$:  $C_3^2:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$ x 4

Degree 9: $C_3^2$, $C_3 \wr C_3 $ x 3

Low degree siblings

9T17 x 3, 27T21, 27T27 x 3

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

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:  $3$
Label:  81.7
magma: IdentifyGroup(G);
 
Character table:

1A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 3E1 3E-1 3F1 3F-1 9A1 9A-1 9B1 9B-1
Size 1 1 1 3 3 3 3 3 3 3 3 9 9 9 9 9 9
3 P 1A 3A-1 3A1 3B1 3E1 3C-1 3C1 3D1 3E-1 3D-1 3B-1 3F-1 3F1 9A-1 9A1 9B-1 9B1
Type
81.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
81.7.1b1 C 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1
81.7.1b2 C 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1
81.7.1c1 C 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1
81.7.1c2 C 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3
81.7.1d1 C 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
81.7.1d2 C 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1
81.7.1e1 C 1 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
81.7.1e2 C 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
81.7.3a1 C 3 3 3 0 0 0 0 3ζ3−1 3ζ3 0 0 0 0 0 0 0 0
81.7.3a2 C 3 3 3 0 0 0 0 3ζ3 3ζ3−1 0 0 0 0 0 0 0 0
81.7.3b1 C 3 3ζ3−1 3ζ3 −1+ζ3 −2−ζ3 1+2ζ3 −1−2ζ3 0 0 1−ζ3 2+ζ3 0 0 0 0 0 0
81.7.3b2 C 3 3ζ3 3ζ3−1 −2−ζ3 −1+ζ3 −1−2ζ3 1+2ζ3 0 0 2+ζ3 1−ζ3 0 0 0 0 0 0
81.7.3c1 C 3 3ζ3−1 3ζ3 −1−2ζ3 1+2ζ3 1−ζ3 2+ζ3 0 0 −2−ζ3 −1+ζ3 0 0 0 0 0 0
81.7.3c2 C 3 3ζ3 3ζ3−1 1+2ζ3 −1−2ζ3 2+ζ3 1−ζ3 0 0 −1+ζ3 −2−ζ3 0 0 0 0 0 0
81.7.3d1 C 3 3ζ3−1 3ζ3 2+ζ3 1−ζ3 −2−ζ3 −1+ζ3 0 0 1+2ζ3 −1−2ζ3 0 0 0 0 0 0
81.7.3d2 C 3 3ζ3 3ζ3−1 1−ζ3 2+ζ3 −1+ζ3 −2−ζ3 0 0 −1−2ζ3 1+2ζ3 0 0 0 0 0 0

magma: CharacterTable(G);