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Properties

Label 27T31
Degree $27$
Order $108$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^3:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(27, 31);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $31$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^3:C_4$
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:  (1,5,8,14)(2,4,9,13)(3,6,7,15)(10,16,25,21)(11,18,26,20)(12,17,27,19)(23,24), (1,21,27,7)(2,20,25,9)(3,19,26,8)(4,5)(10,15,18,23)(11,14,16,22)(12,13,17,24)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$6$:  $S_3$
$12$:  $C_3 : C_4$
$36$:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 9: $C_3^2:C_4$

Low degree siblings

12T72 x 2, 18T54 x 2, 36T89 x 2, 36T94 x 2

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

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:  108.37
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 3B 3C 3D1 3D-1 3E1 3E-1 4A1 4A-1 6A
Size 1 9 2 4 4 4 4 4 4 27 27 18
2 P 1A 1A 3A 3D-1 3B 3C 3E-1 3E1 3D1 2A 2A 3A
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 4A-1 4A1 2A
Type
108.37.1a R 1 1 1 1 1 1 1 1 1 1 1 1
108.37.1b R 1 1 1 1 1 1 1 1 1 −1 −1 1
108.37.1c1 C 1 −1 1 1 1 1 1 1 1 −i i −1
108.37.1c2 C 1 −1 1 1 1 1 1 1 1 i −i −1
108.37.2a R 2 2 −1 2 2 −1 −1 −1 −1 0 0 −1
108.37.2b S 2 −2 −1 2 2 −1 −1 −1 −1 0 0 1
108.37.4a R 4 0 4 −2 1 1 1 −2 −2 0 0 0
108.37.4b R 4 0 4 1 −2 −2 −2 1 1 0 0 0
108.37.4c1 C 4 0 −2 −2 1 −2−3ζ3 1+3ζ3 1 1 0 0 0
108.37.4c2 C 4 0 −2 −2 1 1+3ζ3 −2−3ζ3 1 1 0 0 0
108.37.4d1 C 4 0 −2 1 −2 1 1 −2−3ζ3 1+3ζ3 0 0 0
108.37.4d2 C 4 0 −2 1 −2 1 1 1+3ζ3 −2−3ζ3 0 0 0

magma: CharacterTable(G);