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Properties

Label 18T85
Degree $18$
Order $162$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^3:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(18, 85);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $85$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^3:C_6$
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:  (1,3)(2,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,18), (1,14,8,4,15,9,17,11,5)(2,13,7,3,16,10,18,12,6)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$
$54$:  $C_3^2 : C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 9: $(C_3^3:C_3):C_2$

Low degree siblings

9T22 x 3, 18T85 x 2, 27T53 x 3, 27T62, 27T63

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

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:  162.11
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 3B 3C 3D 3E 3F1 3F-1 6A1 6A-1 9A1 9A-1
Size 1 27 2 6 6 6 6 9 9 27 27 18 18
2 P 1A 1A 3A 3D 3B 3C 3E 3F-1 3F1 3F1 3F-1 9A-1 9A1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 2A 2A 3A 3A
Type
162.11.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
162.11.1b R 1 −1 1 1 1 1 1 1 1 −1 −1 1 1
162.11.1c1 C 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
162.11.1c2 C 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
162.11.1d1 C 1 −1 1 1 1 1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3
162.11.1d2 C 1 −1 1 1 1 1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1
162.11.2a R 2 0 2 −1 −1 −1 2 2 2 0 0 −1 −1
162.11.2b1 C 2 0 2 −1 −1 −1 2 2ζ3−1 2ζ3 0 0 −ζ3−1 −ζ3
162.11.2b2 C 2 0 2 −1 −1 −1 2 2ζ3 2ζ3−1 0 0 −ζ3 −ζ3−1
162.11.6a R 6 0 6 0 0 0 −3 0 0 0 0 0 0
162.11.6b R 6 0 −3 −3 0 3 0 0 0 0 0 0 0
162.11.6c R 6 0 −3 0 3 −3 0 0 0 0 0 0 0
162.11.6d R 6 0 −3 3 −3 0 0 0 0 0 0 0 0

magma: CharacterTable(G);