↑

Properties

Label 18T217
Degree $18$
Order $648$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^3:S_4$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(18, 217);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $217$
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);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2)(3,4)(5,13)(6,14)(7,15,9,11)(8,16,10,12)(17,18), (1,10,15,3,8,11,17,6,13)(2,9,16,4,7,12,18,5,14)
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: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

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

Low degree siblings

9T30, 12T177 x 2, 12T178, 18T218, 18T221, 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

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

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:

1A 2A 2B 3A 3B 3C 3D 4A 6A 6B 9A 9B 12A1 12A5
Size 1 27 54 6 8 12 72 54 54 108 72 72 54 54
2 P 1A 1A 1A 3A 3B 3C 3D 2A 3A 3C 9A 9B 6A 6A
3 P 1A 2A 2B 1A 1A 1A 1A 4A 2A 2B 3B 3B 4A 4A
Type
648.703.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
648.703.1b R 1 1 −1 1 1 1 1 −1 1 −1 1 1 −1 −1
648.703.2a R 2 2 0 2 2 2 −1 0 2 0 −1 −1 0 0
648.703.3a R 3 −1 1 3 3 3 0 −1 −1 1 0 0 −1 −1
648.703.3b R 3 −1 −1 3 3 3 0 1 −1 −1 0 0 1 1
648.703.6a R 6 2 0 3 −3 0 0 2 −1 0 0 0 −1 −1
648.703.6b R 6 2 0 3 −3 0 0 −2 −1 0 0 0 1 1
648.703.6c1 R 6 −2 0 3 −3 0 0 0 1 0 0 0 −ζ12−1−ζ12 ζ12−1+ζ12
648.703.6c2 R 6 −2 0 3 −3 0 0 0 1 0 0 0 ζ12−1+ζ12 −ζ12−1−ζ12
648.703.8a R 8 0 0 −4 −1 2 −1 0 0 0 −1 2 0 0
648.703.8b R 8 0 0 −4 −1 2 −1 0 0 0 2 −1 0 0
648.703.8c R 8 0 0 −4 −1 2 2 0 0 0 −1 −1 0 0
648.703.12a R 12 0 2 0 3 −3 0 0 0 −1 0 0 0 0
648.703.12b R 12 0 −2 0 3 −3 0 0 0 1 0 0 0 0

magma: CharacterTable(G);