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Properties

Label 18T29
Degree $18$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times D_6$

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Show commands: Magma

magma: G := TransitiveGroup(18, 29);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $29$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times D_6$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,9,13,3,7,16)(2,10,14,4,8,15)(5,12,18,6,11,17), (3,17)(4,18)(5,10)(6,9)(11,15)(12,16), (1,2)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,11)(10,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$ x 2
$8$:  $C_2^3$
$12$:  $D_{6}$ x 6
$24$:  $S_3 \times C_2^2$ x 2
$36$:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 2

Degree 6: $D_{6}$ x 2

Degree 9: $S_3^2$

Low degree siblings

12T37 x 2, 18T29 x 3, 24T73, 36T34 x 2, 36T40 x 4

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  72.46
magma: IdentifyGroup(G);
 
Character table:    
      2  3  3  3  3  3  3  3  3  2  2  2  2  1  2  1  2  2  2
      3  2  1  1  .  2  1  1  .  2  1  2  1  2  1  2  1  2  2

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

X.1      1  1  1  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 -1  1  1 -1
X.3      1 -1 -1  1  1 -1 -1  1  1 -1  1 -1  1 -1  1 -1  1  1
X.4      1 -1  1 -1 -1  1 -1  1 -1  1  1 -1  1  1 -1 -1  1 -1
X.5      1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1  1  1  1  1  1
X.6      1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1 -1 -1  1  1 -1
X.7      1  1 -1 -1  1  1 -1 -1  1  1  1  1  1 -1  1 -1  1  1
X.8      1  1  1  1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1  1 -1
X.9      2  . -2  . -2  .  2  . -2  .  2  . -1  1  1 -1 -1  1
X.10     2  . -2  .  2  . -2  .  2  .  2  . -1  1 -1  1 -1 -1
X.11     2  .  2  . -2  . -2  . -2  .  2  . -1 -1  1  1 -1  1
X.12     2  .  2  .  2  .  2  .  2  .  2  . -1 -1 -1 -1 -1 -1
X.13     2 -2  .  . -2  2  .  .  1 -1 -1  1 -1  .  1  .  2 -2
X.14     2 -2  .  .  2 -2  .  . -1  1 -1  1 -1  . -1  .  2  2
X.15     2  2  .  . -2 -2  .  .  1  1 -1 -1 -1  .  1  .  2 -2
X.16     2  2  .  .  2  2  .  . -1 -1 -1 -1 -1  . -1  .  2  2
X.17     4  .  .  . -4  .  .  .  2  . -2  .  1  . -1  . -2  2
X.18     4  .  .  .  4  .  .  . -2  . -2  .  1  .  1  . -2 -2

magma: CharacterTable(G);