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Properties

Label 18T45
Degree $18$
Order $108$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{18}:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(18, 45);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $45$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{18}:C_6$
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:  (3,6,10,17,15,12)(4,5,9,18,16,11)(7,13)(8,14), (1,18,14,5,8,11)(2,17,13,6,7,12)(3,4)(9,15)(10,16)
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
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$36$:  $C_6\times S_3$
$54$:  $(C_9:C_3):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 9: $(C_9:C_3):C_2$

Low degree siblings

18T45, 36T67

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

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:  108.26
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B1 3B-1 6A 6B1 6B-1 6C1 6C-1 6D1 6D-1 9A 9B1 9B-1 18A 18B1 18B-1
Size 1 1 9 9 2 3 3 2 3 3 9 9 9 9 6 6 6 6 6 6
2 P 1A 1A 1A 1A 3A 3B-1 3B1 3A 3B1 3B-1 3B1 3B1 3B-1 3B-1 9B-1 9B1 9A 9A 9B1 9B-1
3 P 1A 2A 2B 2C 1A 1A 1A 2A 2A 2A 2B 2C 2C 2B 3A 3A 3A 6A 6A 6A
Type
108.26.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
108.26.1b R 1 −1 −1 1 1 1 1 −1 −1 −1 −1 −1 1 1 1 1 1 −1 −1 −1
108.26.1c R 1 −1 1 −1 1 1 1 −1 −1 −1 1 1 −1 −1 1 1 1 −1 −1 −1
108.26.1d R 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1
108.26.1e1 C 1 1 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 ζ3−1 ζ3
108.26.1e2 C 1 1 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3−1 ζ3 1 ζ3 ζ3−1
108.26.1f1 C 1 −1 −1 1 1 ζ3−1 ζ3 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1 1 ζ3 ζ3−1 −1 −ζ3−1 −ζ3
108.26.1f2 C 1 −1 −1 1 1 ζ3 ζ3−1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3 1 ζ3−1 ζ3 −1 −ζ3 −ζ3−1
108.26.1g1 C 1 −1 1 −1 1 ζ3−1 ζ3 −1 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1 1 ζ3 ζ3−1 −1 −ζ3−1 −ζ3
108.26.1g2 C 1 −1 1 −1 1 ζ3 ζ3−1 −1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3 1 ζ3−1 ζ3 −1 −ζ3 −ζ3−1
108.26.1h1 C 1 1 −1 −1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 1 ζ3 ζ3−1 1 ζ3−1 ζ3
108.26.1h2 C 1 1 −1 −1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 1 ζ3−1 ζ3 1 ζ3 ζ3−1
108.26.2a R 2 2 0 0 2 2 2 2 2 2 0 0 0 0 −1 −1 −1 −1 −1 −1
108.26.2b R 2 −2 0 0 2 2 2 −2 −2 −2 0 0 0 0 −1 −1 −1 1 1 1
108.26.2c1 C 2 2 0 0 2 2ζ3−1 2ζ3 2 2ζ3 2ζ3−1 0 0 0 0 −1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3
108.26.2c2 C 2 2 0 0 2 2ζ3 2ζ3−1 2 2ζ3−1 2ζ3 0 0 0 0 −1 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1
108.26.2d1 C 2 −2 0 0 2 2ζ3−1 2ζ3 −2 −2ζ3 −2ζ3−1 0 0 0 0 −1 −ζ3 −ζ3−1 1 ζ3−1 ζ3
108.26.2d2 C 2 −2 0 0 2 2ζ3 2ζ3−1 −2 −2ζ3−1 −2ζ3 0 0 0 0 −1 −ζ3−1 −ζ3 1 ζ3 ζ3−1
108.26.6a R 6 6 0 0 −3 0 0 −3 0 0 0 0 0 0 0 0 0 0 0 0
108.26.6b R 6 −6 0 0 −3 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);