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Properties

Label 18T319
Degree $18$
Order $1296$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\wr S_3$

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Show commands: Magma

magma: G := TransitiveGroup(18, 319);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $319$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\wr S_3$
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:  (1,8,13)(2,7,14)(3,10,15)(4,9,16)(5,12,18)(6,11,17), (1,3,17)(2,4,18), (1,3)(2,4), (1,2)(3,4)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16)(17,18)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$24$:  $S_4$
$48$:  $S_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 9: $S_3\wr S_3$

Low degree siblings

9T31, 12T213, 18T300, 18T303, 18T311, 18T312, 18T314, 18T315, 18T320, 24T2893, 24T2894, 24T2895, 24T2912, 27T296, 27T298, 36T2197, 36T2198, 36T2199, 36T2201, 36T2202, 36T2210, 36T2211, 36T2212, 36T2213, 36T2214, 36T2215, 36T2216, 36T2217, 36T2218, 36T2219, 36T2220, 36T2225, 36T2226, 36T2229, 36T2305

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1,17, 3)( 2,18, 4)( 5, 9, 7)( 6,10, 8)(11,15,13)(12,16,14)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $12$ $3$ $( 5, 7, 9)( 6, 8,10)(11,13,15)(12,14,16)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $( 1,17, 3)( 2,18, 4)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $27$ $2$ $( 3,17)( 4,18)(13,15)(14,16)$
$ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ $54$ $6$ $( 1,17)( 2,18)( 5, 9, 7)( 6,10, 8)(11,15)(12,16)$
$ 3, 3, 3, 3, 3, 3 $ $72$ $3$ $( 1, 8,13)( 2, 7,14)( 3,10,15)( 4, 9,16)( 5,12,18)( 6,11,17)$
$ 9, 9 $ $144$ $9$ $( 1,10,11,17, 8,15, 3, 6,13)( 2, 9,12,18, 7,16, 4, 5,14)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $(13,15)(14,16)$
$ 3, 3, 3, 3, 2, 2, 1, 1 $ $36$ $6$ $( 1,17, 3)( 2,18, 4)( 5, 9, 7)( 6,10, 8)(11,15)(12,16)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $36$ $6$ $( 5, 7, 9)( 6, 8,10)(11,13)(12,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $27$ $2$ $( 3,17)( 4,18)( 7, 9)( 8,10)(13,15)(14,16)$
$ 6, 6, 3, 3 $ $216$ $6$ $( 1, 8,13, 3,10,15)( 2, 7,14, 4, 9,16)( 5,12,18)( 6,11,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $18$ $2$ $( 1, 2)( 3, 4)( 5,11)( 6,12)( 7,13)( 8,14)( 9,15)(10,16)(17,18)$
$ 6, 6, 6 $ $72$ $6$ $( 1,18, 3, 2,17, 4)( 5,15, 7,11, 9,13)( 6,16, 8,12,10,14)$
$ 6, 6, 2, 2, 2 $ $36$ $6$ $( 1, 2)( 3, 4)( 5,13, 9,11, 7,15)( 6,14,10,12, 8,16)(17,18)$
$ 6, 2, 2, 2, 2, 2, 2 $ $36$ $6$ $( 1,18, 3, 2,17, 4)( 5,11)( 6,12)( 7,13)( 8,14)( 9,15)(10,16)$
$ 4, 4, 2, 2, 2, 2, 2 $ $162$ $4$ $( 1, 2)( 3,18)( 4,17)( 5,11)( 6,12)( 7,15, 9,13)( 8,16,10,14)$
$ 4, 4, 2, 2, 2, 2, 2 $ $54$ $4$ $( 1, 2)( 3, 4)( 5,11)( 6,12)( 7,15, 9,13)( 8,16,10,14)(17,18)$
$ 6, 4, 4, 2, 2 $ $108$ $12$ $( 1,18, 3, 2,17, 4)( 5,15, 7,13)( 6,16, 8,14)( 9,11)(10,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $54$ $2$ $( 1, 2)( 3,18)( 4,17)( 5,11)( 6,12)( 7,13)( 8,14)( 9,15)(10,16)$
$ 6, 6, 2, 2, 2 $ $108$ $6$ $( 1,18)( 2,17)( 3, 4)( 5,15, 7,11, 9,13)( 6,16, 8,12,10,14)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1296=2^{4} \cdot 3^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1296.3490
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 9A 12A
Size 1 9 18 27 27 54 6 8 12 72 54 162 36 36 36 36 54 72 108 216 144 108
2 P 1A 1A 1A 1A 1A 1A 3A 3B 3C 3D 2C 2C 3C 3C 3A 3A 3A 3B 3C 3D 9A 6E
3 P 1A 2A 2B 2C 2D 2E 1A 1A 1A 1A 4A 4B 2A 2B 2A 2B 2C 2B 2E 2D 3B 4A
Type
1296.3490.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1296.3490.1b R 1 −1 −1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 −1 1 −1 1 1
1296.3490.1c R 1 −1 1 1 −1 −1 1 1 1 1 −1 1 1 −1 1 −1 1 1 −1 −1 1 −1
1296.3490.1d R 1 1 −1 1 1 −1 1 1 1 1 −1 −1 −1 1 −1 1 1 −1 −1 1 1 −1
1296.3490.2a R 2 2 0 2 2 0 2 2 2 −1 0 0 0 2 0 2 2 0 0 −1 −1 0
1296.3490.2b R 2 −2 0 2 −2 0 2 2 2 −1 0 0 0 −2 0 −2 2 0 0 1 −1 0
1296.3490.3a R 3 −1 1 −1 3 1 3 3 3 0 −1 −1 1 −1 1 −1 −1 1 1 0 0 −1
1296.3490.3b R 3 −1 −1 −1 3 −1 3 3 3 0 1 1 −1 −1 −1 −1 −1 −1 −1 0 0 1
1296.3490.3c R 3 1 −1 −1 −3 1 3 3 3 0 −1 1 −1 1 −1 1 −1 −1 1 0 0 −1
1296.3490.3d R 3 1 1 −1 −3 −1 3 3 3 0 1 −1 1 1 1 1 −1 1 −1 0 0 1
1296.3490.6a R 6 4 2 2 0 0 3 −3 0 0 2 0 −1 1 2 −2 −1 −1 0 0 0 −1
1296.3490.6b R 6 −4 −2 2 0 0 3 −3 0 0 2 0 1 −1 −2 2 −1 1 0 0 0 −1
1296.3490.6c R 6 −4 2 2 0 0 3 −3 0 0 −2 0 −1 −1 2 2 −1 −1 0 0 0 1
1296.3490.6d R 6 4 −2 2 0 0 3 −3 0 0 −2 0 1 1 −2 −2 −1 1 0 0 0 1
1296.3490.8a R 8 0 4 0 0 0 −4 −1 2 2 0 0 −2 0 −2 0 0 1 0 0 −1 0
1296.3490.8b R 8 0 −4 0 0 0 −4 −1 2 2 0 0 2 0 2 0 0 −1 0 0 −1 0
1296.3490.12a R 12 4 2 0 0 2 0 3 −3 0 0 0 2 −2 −1 1 0 −1 −1 0 0 0
1296.3490.12b R 12 0 0 −4 0 0 6 −6 0 0 0 0 0 0 0 0 2 0 0 0 0 0
1296.3490.12c R 12 −4 2 0 0 −2 0 3 −3 0 0 0 2 2 −1 −1 0 −1 1 0 0 0
1296.3490.12d R 12 4 −2 0 0 −2 0 3 −3 0 0 0 −2 −2 1 1 0 1 1 0 0 0
1296.3490.12e R 12 −4 −2 0 0 2 0 3 −3 0 0 0 −2 2 1 −1 0 1 −1 0 0 0
1296.3490.16a R 16 0 0 0 0 0 −8 −2 4 −2 0 0 0 0 0 0 0 0 0 0 1 0

magma: CharacterTable(G);