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Properties

Label 18T49
Degree $18$
Order $108$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $\He_3:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(18, 49);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $49$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\He_3:C_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,8,14)(2,9,15)(3,7,13)(4,10,18)(5,11,16)(6,12,17), (1,6,15,10,3,5,14,12,2,4,13,11)(7,17,8,18,9,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$36$:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $C_3^2:C_4$

Degree 9: None

Low degree siblings

18T49, 27T32, 36T85 x 2

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

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:  108.15
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B 3C 4A1 4A-1 6A1 6A-1 12A1 12A-1 12A5 12A-5
Size 1 9 1 1 12 12 9 9 9 9 9 9 9 9
2 P 1A 1A 3A-1 3A1 3B 3C 2A 2A 3A1 3A-1 6A1 6A-1 6A-1 6A1
3 P 1A 2A 1A 1A 1A 1A 4A-1 4A1 2A 2A 4A1 4A-1 4A1 4A-1
Type
108.15.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
108.15.1b R 1 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1
108.15.1c1 C 1 −1 1 1 1 1 −i i −1 −1 i −i i −i
108.15.1c2 C 1 −1 1 1 1 1 i −i −1 −1 −i i −i i
108.15.3a1 C 3 −1 3ζ3−1 3ζ3 0 0 1 1 −ζ3 −ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
108.15.3a2 C 3 −1 3ζ3 3ζ3−1 0 0 1 1 −ζ3−1 −ζ3 ζ3 ζ3−1 ζ3−1 ζ3
108.15.3b1 C 3 −1 3ζ3−1 3ζ3 0 0 −1 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
108.15.3b2 C 3 −1 3ζ3 3ζ3−1 0 0 −1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
108.15.3c1 C 3 1 −3ζ122 3ζ124 0 0 −ζ123 ζ123 ζ124 −ζ122 −ζ125 ζ12 −ζ12 ζ125
108.15.3c2 C 3 1 3ζ124 −3ζ122 0 0 ζ123 −ζ123 −ζ122 ζ124 ζ12 −ζ125 ζ125 −ζ12
108.15.3c3 C 3 1 −3ζ122 3ζ124 0 0 ζ123 −ζ123 ζ124 −ζ122 ζ125 −ζ12 ζ12 −ζ125
108.15.3c4 C 3 1 3ζ124 −3ζ122 0 0 −ζ123 ζ123 −ζ122 ζ124 −ζ12 ζ125 −ζ125 ζ12
108.15.4a R 4 0 4 4 −2 1 0 0 0 0 0 0 0 0
108.15.4b R 4 0 4 4 1 −2 0 0 0 0 0 0 0 0

magma: CharacterTable(G);