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Properties

Label 27T15
Degree $27$
Order $54$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times C_3^2$

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Show commands: Magma

magma: G := TransitiveGroup(27, 15);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $15$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times C_3^2$
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:  (1,11,19)(2,12,20)(3,10,21)(4,9,22,27,15,18)(5,7,23,25,13,16)(6,8,24,26,14,17), (1,27,5)(2,25,6)(3,26,4)(7,14,12)(8,15,10)(9,13,11)(16,24,20)(17,22,21)(18,23,19)
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, $C_6 \times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$ x 4, $S_3$

Degree 9: $C_3^2$, $S_3\times C_3$ x 4

Low degree siblings

18T17 x 4

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $54=2 \cdot 3^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  54.12
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);