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Properties

Label 18T221
Degree $18$
Order $648$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^3:S_4$

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Show commands: Magma

magma: G := TransitiveGroup(18, 221);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $221$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^3:S_4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,6,11)(2,5,12)(3,9,13)(4,10,14)(7,15,17)(8,16,18), (1,10,2,9)(3,8,17,6)(4,7,18,5)(11,15,14)(12,16,13)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$24$:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4$

Degree 9: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$

Low degree siblings

9T30, 12T177 x 2, 12T178, 18T217, 18T218, 18T222, 24T1529 x 2, 24T1530, 27T211, 27T216, 36T1121, 36T1122, 36T1123, 36T1124, 36T1125, 36T1130, 36T1140, 36T1239 x 2, 36T1240

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 $ $1$ $1$ $()$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $( 1,17, 4)( 2,18, 3)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $12$ $3$ $( 1,17, 4)( 2,18, 3)( 5, 8, 9)( 6, 7,10)$
$ 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1,17, 4)( 2,18, 3)( 5, 8, 9)( 6, 7,10)(11,15,14)(12,16,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $27$ $2$ $( 5,10)( 6, 9)( 7, 8)(11,16)(12,15)(13,14)$
$ 3, 3, 2, 2, 2, 2, 2, 2 $ $54$ $6$ $( 1,17, 4)( 2,18, 3)( 5,10)( 6, 9)( 7, 8)(11,16)(12,15)(13,14)$
$ 3, 3, 3, 3, 3, 3 $ $72$ $3$ $( 1, 6,11)( 2, 5,12)( 3, 9,13)( 4,10,14)( 7,15,17)( 8,16,18)$
$ 9, 9 $ $72$ $9$ $( 1, 6,11,17, 7,15, 4,10,14)( 2, 5,12,18, 8,16, 3, 9,13)$
$ 9, 9 $ $72$ $9$ $( 1, 6,11, 4,10,14,17, 7,15)( 2, 5,12, 3, 9,13,18, 8,16)$
$ 4, 4, 4, 1, 1, 1, 1, 1, 1 $ $54$ $4$ $( 5,14, 6,13)( 7,16, 9,11)( 8,15,10,12)$
$ 4, 4, 4, 3, 3 $ $54$ $12$ $( 1,17, 4)( 2,18, 3)( 5,14, 6,13)( 7,16, 9,11)( 8,15,10,12)$
$ 4, 4, 4, 3, 3 $ $54$ $12$ $( 1, 4,17)( 2, 3,18)( 5,14, 6,13)( 7,16, 9,11)( 8,15,10,12)$
$ 6, 6, 2, 2, 2 $ $108$ $6$ $( 1, 3)( 2, 4)( 5,16, 8,13, 9,12)( 6,15, 7,14,10,11)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $54$ $2$ $( 1, 3)( 2, 4)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(17,18)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $648=2^{3} \cdot 3^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  648.703
magma: IdentifyGroup(G);
 
Character table:    
      2  3  2  1  .  3  2  .  .  .  2   2   2  1  2
      3  4  3  3  4  1  1  2  2  2  1   1   1  1  1

        1a 3a 3b 3c 2a 6a 3d 9a 9b 4a 12a 12b 6b 2b
     2P 1a 3a 3b 3c 1a 3a 3d 9a 9b 2a  6a  6a 3b 1a
     3P 1a 1a 1a 1a 2a 2a 1a 3c 3c 4a  4a  4a 2b 2b
     5P 1a 3a 3b 3c 2a 6a 3d 9a 9b 4a 12b 12a 6b 2b
     7P 1a 3a 3b 3c 2a 6a 3d 9a 9b 4a 12b 12a 6b 2b
    11P 1a 3a 3b 3c 2a 6a 3d 9a 9b 4a 12a 12b 6b 2b

X.1      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 -1
X.3      2  2  2  2  2  2 -1 -1 -1  .   .   .  .  .
X.4      3  3  3  3 -1 -1  .  .  . -1  -1  -1  1  1
X.5      3  3  3  3 -1 -1  .  .  .  1   1   1 -1 -1
X.6      6  3  . -3  2 -1  .  .  . -2   1   1  .  .
X.7      6  3  . -3  2 -1  .  .  .  2  -1  -1  .  .
X.8      6  3  . -3 -2  1  .  .  .  .   A  -A  .  .
X.9      6  3  . -3 -2  1  .  .  .  .  -A   A  .  .
X.10     8 -4  2 -1  .  .  2 -1 -1  .   .   .  .  .
X.11     8 -4  2 -1  .  . -1 -1  2  .   .   .  .  .
X.12     8 -4  2 -1  .  . -1  2 -1  .   .   .  .  .
X.13    12  . -3  3  .  .  .  .  .  .   .   . -1  2
X.14    12  . -3  3  .  .  .  .  .  .   .   .  1 -2

A = -E(12)^7+E(12)^11
  = Sqrt(3) = r3

magma: CharacterTable(G);