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Properties

Label 18T23
Degree $18$
Order $54$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(18, 23);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $23$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^2:C_6$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $9$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,18,7,6,14,10)(2,17,8,5,15,12)(3,16,9,4,13,11), (1,8,13)(2,9,14)(3,7,15)(4,5,6)(10,11,12)(16,17,18), (1,9,15)(2,7,13)(3,8,14)(4,6,5)(10,12,11)(16,18,17)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$ x 4, $C_6$
$18$:  $S_3\times C_3$ x 4, $C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$, $S_3\times C_3$ x 3

Degree 9: None

Low degree siblings

18T23 x 3, 27T13

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $54=2 \cdot 3^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  54.13
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B 3C 3D 3E 3F1 3F-1 3G1 3G-1 3H1 3H-1 3I1 3I-1 6A1 6A-1
Size 1 9 1 1 2 2 2 2 2 2 2 2 2 2 2 2 9 9
2 P 1A 1A 3A-1 3A1 3H-1 3F-1 3I-1 3I1 3B 3H1 3E 3F1 3D 3G-1 3C 3G1 3A1 3A-1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A
Type
54.13.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
54.13.1b R 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1
54.13.1c1 C 1 1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
54.13.1c2 C 1 1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
54.13.1d1 C 1 −1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1
54.13.1d2 C 1 −1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3
54.13.2a R 2 0 2 2 −1 −1 −1 2 −1 −1 −1 −1 2 2 −1 −1 0 0
54.13.2b R 2 0 2 2 −1 −1 2 −1 2 2 −1 −1 −1 −1 −1 −1 0 0
54.13.2c R 2 0 2 2 −1 2 −1 −1 −1 −1 −1 −1 −1 −1 2 2 0 0
54.13.2d R 2 0 2 2 2 −1 −1 −1 −1 −1 2 2 −1 −1 −1 −1 0 0
54.13.2e1 C 2 0 2ζ3−1 2ζ3 −1 −1 −1 2 −ζ3 −ζ3−1 −ζ3−1 −ζ3 2ζ3−1 2ζ3 −ζ3−1 −ζ3 0 0
54.13.2e2 C 2 0 2ζ3 2ζ3−1 −1 −1 −1 2 −ζ3−1 −ζ3 −ζ3 −ζ3−1 2ζ3 2ζ3−1 −ζ3 −ζ3−1 0 0
54.13.2f1 C 2 0 2ζ3−1 2ζ3 −1 −1 2 −1 2ζ3 2ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 0 0
54.13.2f2 C 2 0 2ζ3 2ζ3−1 −1 −1 2 −1 2ζ3−1 2ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 0 0
54.13.2g1 C 2 0 2ζ3−1 2ζ3 −1 2 −1 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 2ζ3−1 2ζ3 0 0
54.13.2g2 C 2 0 2ζ3 2ζ3−1 −1 2 −1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 2ζ3 2ζ3−1 0 0
54.13.2h1 C 2 0 2ζ3−1 2ζ3 2 −1 −1 −1 −ζ3 −ζ3−1 2ζ3−1 2ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 0 0
54.13.2h2 C 2 0 2ζ3 2ζ3−1 2 −1 −1 −1 −ζ3−1 −ζ3 2ζ3 2ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 0 0

magma: CharacterTable(G);