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Properties

Label 27T34
Degree $27$
Order $108$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3:S_3^2$

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Show commands: Magma

magma: G := TransitiveGroup(27, 34);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $34$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3:S_3^2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,19,11)(2,20,12)(3,21,10)(4,17,15,26,23,8)(5,18,13,27,24,9)(6,16,14,25,22,7), (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), (1,26,4)(2,25,5,3,27,6)(7,24,10,18,14,20)(8,23,11,17,15,19)(9,22,12,16,13,21)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 5
$12$:  $D_{6}$ x 5
$18$:  $C_3^2:C_2$
$36$:  $S_3^2$ x 4, 18T12

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 5

Degree 9: $C_3^2:C_2$, $S_3^2$ x 4

Low degree siblings

18T58 x 4, 36T91 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 4,26)( 5,27)( 6,25)( 7,14)( 8,15)( 9,13)(16,22)(17,23)(18,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $9$ $2$ $( 2, 3)( 5, 6)( 7,18)( 8,17)( 9,16)(10,20)(11,19)(12,21)(13,22)(14,24)(15,23) (25,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $27$ $2$ $( 2, 3)( 4,26)( 5,25)( 6,27)( 7,24)( 8,23)( 9,22)(10,20)(11,19)(12,21)(13,16) (14,18)(15,17)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$
$ 6, 6, 6, 3, 3, 3 $ $6$ $6$ $( 1, 2, 3)( 4,27, 6,26, 5,25)( 7,15, 9,14, 8,13)(10,11,12)(16,23,18,22,17,24) (19,20,21)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 4,26)( 2, 5,27)( 3, 6,25)( 7,10,14)( 8,11,15)( 9,12,13)(16,21,22) (17,19,23)(18,20,24)$
$ 6, 6, 6, 6, 3 $ $18$ $6$ $( 1, 4,26)( 2, 6,27, 3, 5,25)( 7,20,14,18,10,24)( 8,19,15,17,11,23) ( 9,21,13,16,12,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 5,25)( 2, 6,26)( 3, 4,27)( 7,11,13)( 8,12,14)( 9,10,15)(16,19,24) (17,20,22)(18,21,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 7,24)( 2, 8,22)( 3, 9,23)( 4,10,18)( 5,11,16)( 6,12,17)(13,19,25) (14,20,26)(15,21,27)$
$ 6, 6, 6, 3, 3, 3 $ $6$ $6$ $( 1, 7,20,26,10,18)( 2, 8,21,27,11,16)( 3, 9,19,25,12,17)( 4,14,24)( 5,15,22) ( 6,13,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 8,23)( 2, 9,24)( 3, 7,22)( 4,11,17)( 5,12,18)( 6,10,16)(13,20,27) (14,21,25)(15,19,26)$
$ 6, 6, 6, 3, 3, 3 $ $6$ $6$ $( 1, 8,19,26,11,17)( 2, 9,20,27,12,18)( 3, 7,21,25,10,16)( 4,15,23)( 5,13,24) ( 6,14,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 9,22)( 2, 7,23)( 3, 8,24)( 4,12,16)( 5,10,17)( 6,11,18)(13,21,26) (14,19,27)(15,20,25)$
$ 6, 6, 6, 3, 3, 3 $ $6$ $6$ $( 1, 9,21,26,12,16)( 2, 7,19,27,10,17)( 3, 8,20,25,11,18)( 4,13,22)( 5,14,23) ( 6,15,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,10,20)( 2,11,21)( 3,12,19)( 4,14,24)( 5,15,22)( 6,13,23)( 7,18,26) ( 8,16,27)( 9,17,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,11,19)( 2,12,20)( 3,10,21)( 4,15,23)( 5,13,24)( 6,14,22)( 7,16,25) ( 8,17,26)( 9,18,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,12,21)( 2,10,19)( 3,11,20)( 4,13,22)( 5,14,23)( 6,15,24)( 7,17,27) ( 8,18,25)( 9,16,26)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  108.39
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 3I 6A 6B 6C 6D 6E
Size 1 3 9 27 2 2 2 2 2 4 4 4 4 6 6 6 6 18
2 P 1A 1A 1A 1A 3A 3E 3C 3B 3D 3I 3F 3H 3G 3B 3A 3E 3C 3D
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2B
Type
108.39.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
108.39.1b R 1 −1 −1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1
108.39.1c R 1 −1 1 −1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1
108.39.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1
108.39.2a R 2 0 2 0 2 2 2 −1 2 −1 −1 −1 −1 0 0 0 0 −1
108.39.2b R 2 2 0 0 −1 −1 −1 2 2 2 −1 −1 −1 −1 −1 −1 2 0
108.39.2c R 2 2 0 0 −1 −1 2 2 −1 −1 −1 −1 2 −1 −1 2 −1 0
108.39.2d R 2 2 0 0 −1 2 −1 2 −1 −1 −1 2 −1 −1 2 −1 −1 0
108.39.2e R 2 2 0 0 2 −1 −1 2 −1 −1 2 −1 −1 2 −1 −1 −1 0
108.39.2f R 2 −2 0 0 −1 −1 −1 2 2 2 −1 −1 −1 1 1 1 −2 0
108.39.2g R 2 −2 0 0 −1 −1 2 2 −1 −1 −1 −1 2 1 1 −2 1 0
108.39.2h R 2 −2 0 0 −1 2 −1 2 −1 −1 −1 2 −1 1 −2 1 1 0
108.39.2i R 2 −2 0 0 2 −1 −1 2 −1 −1 2 −1 −1 −2 1 1 1 0
108.39.2j R 2 0 −2 0 2 2 2 −1 2 −1 −1 −1 −1 0 0 0 0 1
108.39.4a R 4 0 0 0 −2 −2 −2 −2 4 −2 1 1 1 0 0 0 0 0
108.39.4b R 4 0 0 0 −2 −2 4 −2 −2 1 1 1 −2 0 0 0 0 0
108.39.4c R 4 0 0 0 −2 4 −2 −2 −2 1 1 −2 1 0 0 0 0 0
108.39.4d R 4 0 0 0 4 −2 −2 −2 −2 1 −2 1 1 0 0 0 0 0

magma: CharacterTable(G);