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Properties

Label 18T57
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, 57);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $57$
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);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,11)(2,12)(3,13)(4,14)(15,17)(16,18), (1,7,14,2,8,13)(3,5,16,17,10,11)(4,6,15,18,9,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$12$:  $D_{6}$ x 2
$36$:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 9: $C_3^2 : D_{6} $

Low degree siblings

9T18 x 2, 18T51 x 2, 18T55 x 2, 18T56, 18T57, 27T29, 36T87 x 2, 36T90

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

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:  108.17
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 3D 6A 6B 6C
Size 1 9 9 9 2 6 6 12 18 18 18
2 P 1A 1A 1A 1A 3A 3B 3C 3D 3B 3C 3A
3 P 1A 2B 2C 2A 1A 1A 1A 1A 2A 2B 2C
Type
108.17.1a R 1 1 1 1 1 1 1 1 1 1 1
108.17.1b R 1 −1 −1 1 1 1 1 1 −1 −1 1
108.17.1c R 1 −1 1 −1 1 1 1 1 −1 1 −1
108.17.1d R 1 1 −1 −1 1 1 1 1 1 −1 −1
108.17.2a R 2 0 2 0 2 2 −1 −1 0 −1 0
108.17.2b R 2 2 0 0 2 −1 2 −1 −1 0 0
108.17.2c R 2 −2 0 0 2 −1 2 −1 1 0 0
108.17.2d R 2 0 −2 0 2 2 −1 −1 0 1 0
108.17.4a R 4 0 0 0 4 −2 −2 1 0 0 0
108.17.6a R 6 0 0 2 −3 0 0 0 0 0 −1
108.17.6b R 6 0 0 −2 −3 0 0 0 0 0 1

magma: CharacterTable(G);