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Properties

Label 18T41
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, 41);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $41$
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:  (1,11,17,13,4,16)(2,12,18,14,3,15)(5,9)(6,10)(7,8), (1,17)(2,18)(5,16,8,13,10,11)(6,15,7,14,9,12)
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: $S_3$

Degree 6: $D_{6}$

Degree 9: $C_3^2 : C_6$

Low degree siblings

18T41, 18T42 x 2, 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 $ $3$ $3$ $( 5, 8,10)( 6, 7, 9)(11,16,13)(12,15,14)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $3$ $3$ $( 5,10, 8)( 6, 9, 7)(11,13,16)(12,14,15)$
$ 6, 6, 2, 2, 1, 1 $ $9$ $6$ $( 3,18)( 4,17)( 5,11, 8,16,10,13)( 6,12, 7,15, 9,14)$
$ 6, 6, 2, 2, 1, 1 $ $9$ $6$ $( 3,18)( 4,17)( 5,13,10,16, 8,11)( 6,14, 9,15, 7,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $9$ $2$ $( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$
$ 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 $ $3$ $6$ $( 1, 2)( 3, 4)( 5, 7,10, 6, 8, 9)(11,15,13,12,16,14)(17,18)$
$ 6, 6, 2, 2, 2 $ $3$ $6$ $( 1, 2)( 3, 4)( 5, 9, 8, 6,10, 7)(11,14,16,12,13,15)(17,18)$
$ 6, 6, 2, 2, 2 $ $9$ $6$ $( 1, 2)( 3,17)( 4,18)( 5,12, 8,15,10,14)( 6,11, 7,16, 9,13)$
$ 6, 6, 2, 2, 2 $ $9$ $6$ $( 1, 2)( 3,17)( 4,18)( 5,14,10,15, 8,12)( 6,13, 9,16, 7,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 2)( 3,17)( 4,18)( 5,15)( 6,16)( 7,13)( 8,14)( 9,11)(10,12)$
$ 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 $ $6$ $6$ $( 1, 5,12, 2, 6,11)( 3, 7,13, 4, 8,14)( 9,16,17,10,15,18)$
$ 6, 6, 6 $ $6$ $6$ $( 1, 5,14, 2, 6,13)( 3, 7,16, 4, 8,15)( 9,11,17,10,12,18)$
$ 6, 6, 6 $ $6$ $6$ $( 1, 5,15, 2, 6,16)( 3, 7,11, 4, 8,12)( 9,13,17,10,14,18)$
$ 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 6,12)( 2, 5,11)( 3, 8,13)( 4, 7,14)( 9,15,17)(10,16,18)$
$ 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 6,14)( 2, 5,13)( 3, 8,16)( 4, 7,15)( 9,12,17)(10,11,18)$
$ 3, 3, 3, 3, 3, 3 $ $6$ $3$ $( 1, 6,15)( 2, 5,16)( 3, 8,11)( 4, 7,12)( 9,14,17)(10,13,18)$

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);