# Properties

 Label 18T1 Degree $18$ Order $18$ Cyclic yes Abelian yes Solvable yes Primitive no $p$-group no Group: $C_{18}$

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Show commands: Magma

magma: G := TransitiveGroup(18, 1);

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$9$:  $C_9$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 9: $C_9$

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $18=2 \cdot 3^{2}$ magma: Order(G); Cyclic: yes magma: IsCyclic(G); Abelian: yes magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: $1$ Label: 18.2 magma: IdentifyGroup(G);
 Character table:  2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1a 2a 9a 18a 18b 9b 3a 6a 18c 9c 9d 18d 6b 3b 18e 9e 18f 9f X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 X.3 1 1 A A /A /A 1 1 A A /A /A 1 1 A A /A /A X.4 1 -1 A -A -/A /A 1 -1 -A A /A -/A -1 1 -A A -/A /A X.5 1 1 /A /A A A 1 1 /A /A A A 1 1 /A /A A A X.6 1 -1 /A -/A -A A 1 -1 -/A /A A -A -1 1 -/A /A -A A X.7 1 1 B B /D /D A A C C /C /C /A /A D D /B /B X.8 1 -1 B -B -/D /D A -A -C C /C -/C -/A /A -D D -/B /B X.9 1 1 C C /B /B A A D D /D /D /A /A B B /C /C X.10 1 -1 C -C -/B /B A -A -D D /D -/D -/A /A -B B -/C /C X.11 1 1 D D /C /C A A B B /B /B /A /A C C /D /D X.12 1 -1 D -D -/C /C A -A -B B /B -/B -/A /A -C C -/D /D X.13 1 1 /D /D C C /A /A /B /B B B A A /C /C D D X.14 1 -1 /D -/D -C C /A -/A -/B /B B -B -A A -/C /C -D D X.15 1 1 /C /C B B /A /A /D /D D D A A /B /B C C X.16 1 -1 /C -/C -B B /A -/A -/D /D D -D -A A -/B /B -C C X.17 1 1 /B /B D D /A /A /C /C C C A A /D /D B B X.18 1 -1 /B -/B -D D /A -/A -/C /C C -C -A A -/D /D -B B A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(9)^2 C = -E(9)^2-E(9)^5 D = E(9)^5 

magma: CharacterTable(G);