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Properties

Label 27T43
Degree $27$
Order $162$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(C_3\times C_9).S_3$

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Show commands: Magma

magma: G := TransitiveGroup(27, 43);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $43$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $(C_3\times C_9).S_3$
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:  (1,21,17,2,19,18,3,20,16)(4,22,11,5,23,12,6,24,10)(7,26,13,8,27,14,9,25,15), (1,23,15,2,24,13,3,22,14)(4,26,18,5,27,16,6,25,17)(7,21,10,8,19,11,9,20,12), (1,27,2,26,3,25)(4,24,5,23,6,22)(7,19,8,21,9,20)(10,12)(13,16)(14,18)(15,17)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 4

Degree 9: $C_3^2:C_2$

Low degree siblings

There are no siblings 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 $ $3$ $3$ $(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $3$ $(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,26,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $27$ $2$ $( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,25)(11,27)(12,26)(13,23)(14,22)(15,24)(16,19) (17,21)(18,20)$
$ 6, 6, 6, 2, 2, 2, 2, 1 $ $27$ $6$ $( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,26,11,25,12,27)(13,24,14,23,15,22) (16,20,17,19,18,21)$
$ 6, 6, 6, 2, 2, 2, 2, 1 $ $27$ $6$ $( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,27,12,25,11,26)(13,22,15,23,14,24) (16,21,18,19,17,20)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$
$ 9, 9, 9 $ $6$ $9$ $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,13,18,11,14,16,12,15,17)(19,23,26,20,24,27,21, 22,25)$
$ 9, 9, 9 $ $6$ $9$ $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,15,16,11,13,17,12,14,18)(19,24,25,20,22,26,21, 23,27)$
$ 9, 9, 9 $ $6$ $9$ $( 1, 5, 8, 2, 6, 9, 3, 4, 7)(10,13,18,11,14,16,12,15,17)(19,22,27,20,23,25,21, 24,26)$
$ 9, 9, 9 $ $18$ $9$ $( 1,10,25, 3,12,27, 2,11,26)( 4,14,19, 6,13,21, 5,15,20)( 7,18,24, 9,17,23, 8, 16,22)$
$ 9, 9, 9 $ $18$ $9$ $( 1,13,22, 3,15,24, 2,14,23)( 4,16,25, 6,18,27, 5,17,26)( 7,11,20, 9,10,19, 8, 12,21)$
$ 9, 9, 9 $ $18$ $9$ $( 1,16,21, 3,18,20, 2,17,19)( 4,10,22, 6,12,24, 5,11,23)( 7,15,26, 9,14,25, 8, 13,27)$

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:  162.22
magma: IdentifyGroup(G);
 
Character table:    
      2  1  1  1  1   1   1  .  .  .  .  .  .  .
      3  4  3  3  1   1   1  4  3  3  3  2  2  2

        1a 3a 3b 2a  6a  6b 3c 9a 9b 9c 9d 9e 9f
     2P 1a 3b 3a 1a  3a  3b 3c 9c 9a 9b 9d 9e 9f
     3P 1a 1a 1a 2a  2a  2a 1a 3c 3c 3c 3c 3c 3c
     5P 1a 3b 3a 2a  6b  6a 3c 9b 9c 9a 9d 9e 9f
     7P 1a 3a 3b 2a  6a  6b 3c 9c 9a 9b 9d 9e 9f

X.1      1  1  1  1   1   1  1  1  1  1  1  1  1
X.2      1  1  1 -1  -1  -1  1  1  1  1  1  1  1
X.3      2  2  2  .   .   .  2  2  2  2 -1 -1 -1
X.4      2  2  2  .   .   .  2 -1 -1 -1  2 -1 -1
X.5      2  2  2  .   .   .  2 -1 -1 -1 -1 -1  2
X.6      2  2  2  .   .   .  2 -1 -1 -1 -1  2 -1
X.7      3  A /A -1   B  /B  3  .  .  .  .  .  .
X.8      3 /A  A -1  /B   B  3  .  .  .  .  .  .
X.9      3  A /A  1  -B -/B  3  .  .  .  .  .  .
X.10     3 /A  A  1 -/B  -B  3  .  .  .  .  .  .
X.11     6  .  .  .   .   . -3  C  E  D  .  .  .
X.12     6  .  .  .   .   . -3  D  C  E  .  .  .
X.13     6  .  .  .   .   . -3  E  D  C  .  .  .

A = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
B = -E(3)
  = (1-Sqrt(-3))/2 = -b3
C = 2*E(9)^2+E(9)^4+E(9)^5+2*E(9)^7
D = -E(9)^2+E(9)^4+E(9)^5-E(9)^7
E = -E(9)^2-2*E(9)^4-2*E(9)^5-E(9)^7

magma: CharacterTable(G);