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Properties

Label 27T46
Degree $27$
Order $162$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^3:S_3$

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Show commands: Magma

magma: G := TransitiveGroup(27, 46);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $46$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^3:S_3$
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:  (1,20,12)(2,21,10)(3,19,11)(4,24,14)(5,22,15)(6,23,13)(7,26,18)(8,27,16)(9,25,17), (1,2,3)(4,5,6)(7,16,9,18,8,17)(10,19,12,21,11,20)(13,24,15,23,14,22)(25,26,27), (7,10,13)(8,11,14)(9,12,15)(16,24,19)(17,22,20)(18,23,21)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$ x 4, $C_6$
$18$:  $S_3\times C_3$ x 4, $C_3^2:C_2$
$54$:  $(C_3^2:C_3):C_2$ x 3, 18T23

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$, $S_3$

Degree 9: $S_3\times C_3$, $(C_3^2:C_3):C_2$ x 3

Low degree siblings

27T46 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 $ $6$ $3$ $( 7,10,13)( 8,11,14)( 9,12,15)(16,24,19)(17,22,20)(18,23,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 7,18)( 8,16)( 9,17)(10,21)(11,19)(12,20)(13,23)(14,24)(15,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7,11,15)( 8,12,13)( 9,10,14)(16,22,21)(17,23,19) (18,24,20)(25,26,27)$
$ 6, 6, 6, 3, 3, 3 $ $9$ $6$ $( 1, 2, 3)( 4, 5, 6)( 7,16, 9,18, 8,17)(10,19,12,21,11,20)(13,24,15,23,14,22) (25,26,27)$
$ 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 $ $6$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7,12,14)( 8,10,15)( 9,11,13)(16,23,20)(17,24,21) (18,22,19)(25,27,26)$
$ 6, 6, 6, 3, 3, 3 $ $9$ $6$ $( 1, 3, 2)( 4, 6, 5)( 7,17, 8,18, 9,16)(10,20,11,21,12,19)(13,22,14,23,15,24) (25,27,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 4,26)( 2, 5,27)( 3, 6,25)( 7,12,14)( 8,10,15)( 9,11,13)(16,21,22) (17,19,23)(18,20,24)$
$ 6, 6, 6, 3, 3, 3 $ $9$ $6$ $( 1, 4,26)( 2, 5,27)( 3, 6,25)( 7,17,14,23,12,19)( 8,18,15,24,10,20) ( 9,16,13,22,11,21)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 5,25)( 2, 6,26)( 3, 4,27)( 7,10,13)( 8,11,14)( 9,12,15)(16,19,24) (17,20,22)(18,21,23)$
$ 6, 6, 6, 3, 3, 3 $ $9$ $6$ $( 1, 5,25)( 2, 6,26)( 3, 4,27)( 7,18,13,23,10,21)( 8,16,14,24,11,19) ( 9,17,15,22,12,20)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 6,27)( 2, 4,25)( 3, 5,26)( 7,11,15)( 8,12,13)( 9,10,14)(16,20,23) (17,21,24)(18,19,22)$
$ 6, 6, 6, 3, 3, 3 $ $9$ $6$ $( 1, 6,27)( 2, 4,25)( 3, 5,26)( 7,16,15,23,11,20)( 8,17,13,24,12,21) ( 9,18,14,22,10,19)$
$ 6, 6, 6, 3, 3, 3 $ $9$ $6$ $( 1, 7, 4,12,26,14)( 2, 8, 5,10,27,15)( 3, 9, 6,11,25,13)(16,22,21)(17,23,19) (18,24,20)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 7,16)( 2, 8,17)( 3, 9,18)( 4,12,21)( 5,10,19)( 6,11,20)(13,24,25) (14,22,26)(15,23,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 7,19)( 2, 8,20)( 3, 9,21)( 4,12,23)( 5,10,24)( 6,11,22)(13,16,25) (14,17,26)(15,18,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 7,24)( 2, 8,22)( 3, 9,23)( 4,12,18)( 5,10,16)( 6,11,17)(13,19,25) (14,20,26)(15,21,27)$
$ 6, 6, 6, 3, 3, 3 $ $9$ $6$ $( 1, 8, 6,12,27,13)( 2, 9, 4,10,25,14)( 3, 7, 5,11,26,15)(16,23,20)(17,24,21) (18,22,19)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 8,18)( 2, 9,16)( 3, 7,17)( 4,10,20)( 5,11,21)( 6,12,19)(13,22,27) (14,23,25)(15,24,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 8,21)( 2, 9,19)( 3, 7,20)( 4,10,22)( 5,11,23)( 6,12,24)(13,17,27) (14,18,25)(15,16,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 8,23)( 2, 9,24)( 3, 7,22)( 4,10,17)( 5,11,18)( 6,12,16)(13,20,27) (14,21,25)(15,19,26)$
$ 6, 6, 6, 3, 3, 3 $ $9$ $6$ $( 1, 9, 5,12,25,15)( 2, 7, 6,10,26,13)( 3, 8, 4,11,27,14)(16,24,19)(17,22,20) (18,23,21)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 9,17)( 2, 7,18)( 3, 8,16)( 4,11,19)( 5,12,20)( 6,10,21)(13,23,26) (14,24,27)(15,22,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 9,20)( 2, 7,21)( 3, 8,19)( 4,11,24)( 5,12,22)( 6,10,23)(13,18,26) (14,16,27)(15,17,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 9,22)( 2, 7,23)( 3, 8,24)( 4,11,16)( 5,12,17)( 6,10,18)(13,21,26) (14,19,27)(15,20,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,25, 5)( 2,26, 6)( 3,27, 4)( 7,13,10)( 8,14,11)( 9,15,12)(16,24,19) (17,22,20)(18,23,21)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,26, 4)( 2,27, 5)( 3,25, 6)( 7,14,12)( 8,15,10)( 9,13,11)(16,22,21) (17,23,19)(18,24,20)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,27, 6)( 2,25, 4)( 3,26, 5)( 7,15,11)( 8,13,12)( 9,14,10)(16,23,20) (17,24,21)(18,22,19)$

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:  162.41
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 3E 3F 3G 3H 3I1 3I-1 3J1 3J-1 3K1 3K-1 3L1 3L-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 6D1 6D-1
Size 1 9 1 1 1 1 1 1 1 1 6 6 6 6 6 6 6 6 6 6 6 6 9 9 9 9 9 9 9 9
2 P 1A 1A 3D1 3A1 3C-1 3B1 3B-1 3C1 3D-1 3A-1 3J-1 3H 3K1 3I-1 3L-1 3K-1 3J1 3E 3G 3F 3L1 3I1 3C1 3B1 3D-1 3C-1 3A1 3B-1 3D1 3A-1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2A 2A
Type
162.41.1a R 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 1
162.41.1b R 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 −1
162.41.1c1 C 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1
162.41.1c2 C 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3
162.41.1d1 C 1 −1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −1 −1 −ζ3 −ζ3−1
162.41.1d2 C 1 −1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −1 −1 −ζ3−1 −ζ3
162.41.2a R 2 0 2 2 2 2 2 2 2 2 −1 −1 −1 2 −1 −1 2 2 −1 −1 −1 −1 0 0 0 0 0 0 0 0
162.41.2b R 2 0 2 2 2 2 2 2 2 2 −1 −1 2 −1 −1 −1 −1 −1 2 2 −1 −1 0 0 0 0 0 0 0 0
162.41.2c R 2 0 2 2 2 2 2 2 2 2 −1 2 −1 −1 −1 −1 −1 −1 −1 −1 2 2 0 0 0 0 0 0 0 0
162.41.2d R 2 0 2 2 2 2 2 2 2 2 2 −1 −1 −1 2 2 −1 −1 −1 −1 −1 −1 0 0 0 0 0 0 0 0
162.41.2e1 C 2 0 2ζ3−1 2ζ3 2ζ3−1 2ζ3 2 2 2ζ3−1 2ζ3 −1 −1 −1 2 −ζ3−1 −ζ3 2ζ3−1 2ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 0 0 0 0 0 0 0 0
162.41.2e2 C 2 0 2ζ3 2ζ3−1 2ζ3 2ζ3−1 2 2 2ζ3 2ζ3−1 −1 −1 −1 2 −ζ3 −ζ3−1 2ζ3 2ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 0 0 0 0 0 0 0 0
162.41.2f1 C 2 0 2ζ3−1 2ζ3 2ζ3−1 2ζ3 2 2 2ζ3−1 2ζ3 −1 −1 2 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 2ζ3 2ζ3−1 −ζ3 −ζ3−1 0 0 0 0 0 0 0 0
162.41.2f2 C 2 0 2ζ3 2ζ3−1 2ζ3 2ζ3−1 2 2 2ζ3 2ζ3−1 −1 −1 2 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 2ζ3−1 2ζ3 −ζ3−1 −ζ3 0 0 0 0 0 0 0 0
162.41.2g1 C 2 0 2ζ3−1 2ζ3 2ζ3−1 2ζ3 2 2 2ζ3−1 2ζ3 −1 2 −1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 2ζ3 2ζ3−1 0 0 0 0 0 0 0 0
162.41.2g2 C 2 0 2ζ3 2ζ3−1 2ζ3 2ζ3−1 2 2 2ζ3 2ζ3−1 −1 2 −1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 2ζ3−1 2ζ3 0 0 0 0 0 0 0 0
162.41.2h1 C 2 0 2ζ3−1 2ζ3 2ζ3−1 2ζ3 2 2 2ζ3−1 2ζ3 2 −1 −1 −1 2ζ3−1 2ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 0 0 0 0 0 0 0 0
162.41.2h2 C 2 0 2ζ3 2ζ3−1 2ζ3 2ζ3−1 2 2 2ζ3 2ζ3−1 2 −1 −1 −1 2ζ3 2ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 0 0 0 0 0 0 0 0
162.41.3a1 C 3 1 3ζ3−1 3ζ3 3ζ3 3ζ3−1 3ζ3 3ζ3−1 3 3 0 0 0 0 0 0 0 0 0 0 0 0 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1
162.41.3a2 C 3 1 3ζ3 3ζ3−1 3ζ3−1 3ζ3 3ζ3−1 3ζ3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1
162.41.3b1 C 3 1 3ζ3−1 3ζ3 3 3 3ζ3−1 3ζ3 3ζ3 3ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
162.41.3b2 C 3 1 3ζ3 3ζ3−1 3 3 3ζ3 3ζ3−1 3ζ3−1 3ζ3 0 0 0 0 0 0 0 0 0 0 0 0 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1
162.41.3c1 C 3 1 3 3 3ζ3−1 3ζ3 3ζ3 3ζ3−1 3ζ3 3ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
162.41.3c2 C 3 1 3 3 3ζ3 3ζ3−1 3ζ3−1 3ζ3 3ζ3−1 3ζ3 0 0 0 0 0 0 0 0 0 0 0 0 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
162.41.3d1 C 3 −1 3ζ3−1 3ζ3 3ζ3 3ζ3−1 3ζ3 3ζ3−1 3 3 0 0 0 0 0 0 0 0 0 0 0 0 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −1 −1
162.41.3d2 C 3 −1 3ζ3 3ζ3−1 3ζ3−1 3ζ3 3ζ3−1 3ζ3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −1 −1
162.41.3e1 C 3 −1 3ζ3−1 3ζ3 3 3 3ζ3−1 3ζ3 3ζ3 3ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 −ζ3 −ζ3−1 −1 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
162.41.3e2 C 3 −1 3ζ3 3ζ3−1 3 3 3ζ3 3ζ3−1 3ζ3−1 3ζ3 0 0 0 0 0 0 0 0 0 0 0 0 −ζ3−1 −ζ3 −1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
162.41.3f1 C 3 −1 3 3 3ζ3−1 3ζ3 3ζ3 3ζ3−1 3ζ3 3ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3
162.41.3f2 C 3 −1 3 3 3ζ3 3ζ3−1 3ζ3−1 3ζ3 3ζ3−1 3ζ3 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1

magma: CharacterTable(G);