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Properties

Label 18T261
Degree $18$
Order $1080$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_3\times A_6$

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Show commands: Magma

magma: G := TransitiveGroup(18, 261);
 

Group action invariants

Degree $n$:  $18$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $261$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times A_6$
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:  (1,12,18,14,8,2,10,16,15,9,3,11,17,13,7)(4,6,5), (1,5,11,14,16,2,6,12,15,17,3,4,10,13,18)(7,9,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$360$:  $A_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_6$

Degree 9: None

Low degree siblings

18T261, 30T223, 45T149 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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $45$ $2$ $( 1, 6)( 2, 4)( 3, 5)( 7,11)( 8,12)( 9,10)$
$ 6, 6, 3, 3 $ $45$ $6$ $( 1, 5, 2, 6, 3, 4)( 7,10, 8,11, 9,12)(13,15,14)(16,18,17)$
$ 6, 6, 3, 3 $ $45$ $6$ $( 1, 4, 3, 6, 2, 5)( 7,12, 9,11, 8,10)(13,14,15)(16,17,18)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $40$ $3$ $(1,6,9)(2,4,7)(3,5,8)$
$ 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 5, 7)( 2, 6, 8)( 3, 4, 9)(10,12,11)(13,15,14)(16,18,17)$
$ 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 4, 8)( 2, 5, 9)( 3, 6, 7)(10,11,12)(13,14,15)(16,17,18)$
$ 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 6, 9)( 2, 4, 7)( 3, 5, 8)(10,14,17)(11,15,18)(12,13,16)$
$ 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 5, 7)( 2, 6, 8)( 3, 4, 9)(10,13,18)(11,14,16)(12,15,17)$
$ 3, 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 4, 8)( 2, 5, 9)( 3, 6, 7)(10,15,16)(11,13,17)(12,14,18)$
$ 4, 4, 4, 2, 2, 2 $ $90$ $4$ $( 1, 6, 9,10)( 2, 4, 7,11)( 3, 5, 8,12)(13,16)(14,17)(15,18)$
$ 12, 6 $ $90$ $12$ $( 1, 5, 7,10, 3, 4, 9,12, 2, 6, 8,11)(13,18,14,16,15,17)$
$ 12, 6 $ $90$ $12$ $( 1, 4, 8,10, 2, 5, 9,11, 3, 6, 7,12)(13,17,15,16,14,18)$
$ 5, 5, 5, 1, 1, 1 $ $72$ $5$ $( 1, 6, 9,10,14)( 2, 4, 7,11,15)( 3, 5, 8,12,13)$
$ 15, 3 $ $72$ $15$ $( 1, 5, 7,10,13, 2, 6, 8,11,14, 3, 4, 9,12,15)(16,18,17)$
$ 15, 3 $ $72$ $15$ $( 1, 4, 8,10,15, 3, 6, 7,12,14, 2, 5, 9,11,13)(16,17,18)$
$ 5, 5, 5, 1, 1, 1 $ $72$ $5$ $( 1, 6, 9,10,17)( 2, 4, 7,11,18)( 3, 5, 8,12,16)$
$ 15, 3 $ $72$ $15$ $( 1, 5, 7,10,16, 2, 6, 8,11,17, 3, 4, 9,12,18)(13,15,14)$
$ 15, 3 $ $72$ $15$ $( 1, 4, 8,10,18, 3, 6, 7,12,17, 2, 5, 9,11,16)(13,14,15)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1080=2^{3} \cdot 3^{3} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1080.487
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B 3C 3D1 3D-1 3E1 3E-1 4A 5A1 5A2 6A1 6A-1 12A1 12A-1 15A1 15A-1 15A2 15A-2
Size 1 45 1 1 40 40 40 40 40 40 90 72 72 45 45 90 90 72 72 72 72
2 P 1A 1A 3A-1 3A1 3D-1 3D1 3E1 3B 3E-1 3C 2A 5A2 5A1 3A-1 3A1 6A1 6A-1 15A-1 15A2 15A1 15A-2
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 4A 5A2 5A1 2A 2A 4A 4A 5A2 5A1 5A2 5A1
5 P 1A 2A 3A-1 3A1 3D-1 3D1 3E1 3B 3E-1 3C 4A 1A 1A 6A-1 6A1 12A-1 12A1 3A1 3A1 3A-1 3A-1
Type
1080.487.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1080.487.1b1 C 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
1080.487.1b2 C 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
1080.487.5a R 5 1 5 5 −1 2 −1 −1 2 2 −1 0 0 1 1 −1 −1 0 0 0 0
1080.487.5b R 5 1 5 5 2 −1 2 2 −1 −1 −1 0 0 1 1 −1 −1 0 0 0 0
1080.487.5c1 C 5 1 5ζ3−1 5ζ3 −1 2 −ζ3−1 −ζ3 2ζ3 2ζ3−1 −1 0 0 ζ3−1 ζ3 −ζ3 −ζ3−1 0 0 0 0
1080.487.5c2 C 5 1 5ζ3 5ζ3−1 −1 2 −ζ3 −ζ3−1 2ζ3−1 2ζ3 −1 0 0 ζ3 ζ3−1 −ζ3−1 −ζ3 0 0 0 0
1080.487.5d1 C 5 1 5ζ3−1 5ζ3 2 −1 2ζ3−1 2ζ3 −ζ3 −ζ3−1 −1 0 0 ζ3−1 ζ3 −ζ3 −ζ3−1 0 0 0 0
1080.487.5d2 C 5 1 5ζ3 5ζ3−1 2 −1 2ζ3 2ζ3−1 −ζ3−1 −ζ3 −1 0 0 ζ3 ζ3−1 −ζ3−1 −ζ3 0 0 0 0
1080.487.8a1 R 8 0 8 8 −1 −1 −1 −1 −1 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0 0 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
1080.487.8a2 R 8 0 8 8 −1 −1 −1 −1 −1 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0 0 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
1080.487.8b1 C 8 0 8ζ15−5 8ζ155 −1 −1 −ζ15−5 −ζ155 −ζ155 −ζ15−5 0 −ζ15−3−ζ153 −ζ15−6−ζ156 0 0 0 0 −1+ζ15+ζ152−ζ153+ζ154+ζ157 −ζ15−ζ154 −1+ζ15+ζ154−ζ155 1−ζ15−ζ152+ζ153−ζ154+ζ155−ζ157
1080.487.8b2 C 8 0 8ζ155 8ζ15−5 −1 −1 −ζ155 −ζ15−5 −ζ15−5 −ζ155 0 −ζ15−3−ζ153 −ζ15−6−ζ156 0 0 0 0 −ζ15−ζ154 −1+ζ15+ζ152−ζ153+ζ154+ζ157 1−ζ15−ζ152+ζ153−ζ154+ζ155−ζ157 −1+ζ15+ζ154−ζ155
1080.487.8b3 C 8 0 8ζ15−5 8ζ155 −1 −1 −ζ15−5 −ζ155 −ζ155 −ζ15−5 0 −ζ15−6−ζ156 −ζ15−3−ζ153 0 0 0 0 1−ζ15−ζ152+ζ153−ζ154+ζ155−ζ157 −1+ζ15+ζ154−ζ155 −ζ15−ζ154 −1+ζ15+ζ152−ζ153+ζ154+ζ157
1080.487.8b4 C 8 0 8ζ155 8ζ15−5 −1 −1 −ζ155 −ζ15−5 −ζ15−5 −ζ155 0 −ζ15−6−ζ156 −ζ15−3−ζ153 0 0 0 0 −1+ζ15+ζ154−ζ155 1−ζ15−ζ152+ζ153−ζ154+ζ155−ζ157 −1+ζ15+ζ152−ζ153+ζ154+ζ157 −ζ15−ζ154
1080.487.9a R 9 1 9 9 0 0 0 0 0 0 1 −1 −1 1 1 1 1 −1 −1 −1 −1
1080.487.9b1 C 9 1 9ζ3−1 9ζ3 0 0 0 0 0 0 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
1080.487.9b2 C 9 1 9ζ3 9ζ3−1 0 0 0 0 0 0 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1
1080.487.10a R 10 −2 10 10 1 1 1 1 1 1 0 0 0 −2 −2 0 0 0 0 0 0
1080.487.10b1 C 10 −2 10ζ3−1 10ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 0 0 0 −2ζ3−1 −2ζ3 0 0 0 0 0 0
1080.487.10b2 C 10 −2 10ζ3 10ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 0 0 0 −2ζ3 −2ζ3−1 0 0 0 0 0 0

magma: CharacterTable(G);