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Properties

Label 27T48
Degree $27$
Order $162$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^3:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(27, 48);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $48$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^3:C_6$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3)(4,5,6)(7,11,15)(8,12,13)(9,10,14)(16,24,20)(17,22,21)(18,23,19)(25,26,27), (1,8,24)(2,9,22)(3,7,23)(4,15,18,27,12,20)(5,13,16,25,10,21)(6,14,17,26,11,19)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$ x 4
$6$:  $S_3$, $C_6$ x 4
$9$:  $C_3^2$
$18$:  $S_3\times C_3$ x 4, $C_6 \times C_3$
$54$:  $C_3^2 : C_6$, $C_3^2\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$ x 4

Degree 9: $C_3^2$, $C_3^2 : S_3 $

Low degree siblings

18T76, 18T78, 18T81 x 2, 27T60 x 3

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $162=2 \cdot 3^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  162.34
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);