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Properties

Label 18T42
Degree $18$
Order $108$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2:D_6$

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Show commands: Magma

magma: G := TransitiveGroup(18, 42);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $42$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^2:D_6$
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:  (1,2)(3,4)(5,7,10,6,8,9)(11,15,13,12,16,14)(17,18), (1,5,16,2,6,15)(3,9,12,17,8,13)(4,10,11,18,7,14)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$36$:  $C_6\times S_3$
$54$:  $C_3^2 : C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 9: $C_3^2 : S_3 $

Low degree siblings

18T41 x 2, 18T42, 36T71, 36T73, 36T75

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  108.25
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B1 3B-1 3C 3D1 3D-1 6A 6B1 6B-1 6C 6D1 6D-1 6E1 6E-1 6F1 6F-1
Size 1 1 9 9 2 3 3 6 6 6 2 3 3 6 6 6 9 9 9 9
2 P 1A 1A 1A 1A 3A 3B-1 3B1 3D-1 3D1 3C 3A 3B-1 3B1 3D1 3D-1 3C 3B1 3B-1 3B1 3B-1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2B 2B 2C 2C
Type
108.25.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
108.25.1b R 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1
108.25.1c R 1 −1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 −1 −1
108.25.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
108.25.1e1 C 1 1 1 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1
108.25.1e2 C 1 1 1 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3
108.25.1f1 C 1 −1 −1 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 −1 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1
108.25.1f2 C 1 −1 −1 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 −1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3
108.25.1g1 C 1 −1 1 −1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 −1 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1
108.25.1g2 C 1 −1 1 −1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 −1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3
108.25.1h1 C 1 1 −1 −1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1
108.25.1h2 C 1 1 −1 −1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3
108.25.2a R 2 2 0 0 2 2 2 −1 −1 −1 2 2 2 −1 −1 −1 0 0 0 0
108.25.2b R 2 −2 0 0 2 2 2 −1 −1 −1 −2 −2 −2 1 1 1 0 0 0 0
108.25.2c1 C 2 2 0 0 2 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 2 2ζ3−1 2ζ3 −1 −ζ3 −ζ3−1 0 0 0 0
108.25.2c2 C 2 2 0 0 2 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 2 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 0 0 0 0
108.25.2d1 C 2 −2 0 0 2 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 −2 −2ζ3−1 −2ζ3 1 ζ3 ζ3−1 0 0 0 0
108.25.2d2 C 2 −2 0 0 2 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 −2 −2ζ3 −2ζ3−1 1 ζ3−1 ζ3 0 0 0 0
108.25.6a R 6 6 0 0 −3 0 0 0 0 0 −3 0 0 0 0 0 0 0 0 0
108.25.6b R 6 −6 0 0 −3 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);