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Properties

Label 27T9
Degree $27$
Order $54$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times D_9$

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Show commands: Magma

magma: G := TransitiveGroup(27, 9);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times D_9$
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:  (1,22,17,11,6,25,20,13,7)(2,23,18,12,4,26,21,14,8)(3,24,16,10,5,27,19,15,9), (1,2,3)(4,27,6,26,5,25)(7,23,9,22,8,24)(10,20,12,19,11,21)(13,18,15,17,14,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$, $D_{9}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$, $S_3$

Degree 9: $D_{9}$, $S_3\times C_3$

Low degree siblings

18T19

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, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $9$ $2$ $( 4,26)( 5,27)( 6,25)( 7,22)( 8,23)( 9,24)(10,19)(11,20)(12,21)(13,17)(14,18) (15,16)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$
$ 6, 6, 6, 6, 3 $ $9$ $6$ $( 1, 2, 3)( 4,27, 6,26, 5,25)( 7,23, 9,22, 8,24)(10,20,12,19,11,21) (13,18,15,17,14,16)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$
$ 6, 6, 6, 6, 3 $ $9$ $6$ $( 1, 3, 2)( 4,25, 5,26, 6,27)( 7,24, 8,22, 9,23)(10,21,11,19,12,20) (13,16,14,17,15,18)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 4, 9,11,14,16,20,23,27)( 2, 5, 7,12,15,17,21,24,25)( 3, 6, 8,10,13,18,19, 22,26)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 5, 8,11,15,18,20,24,26)( 2, 6, 9,12,13,16,21,22,27)( 3, 4, 7,10,14,17,19, 23,25)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 6, 7,11,13,17,20,22,25)( 2, 4, 8,12,14,18,21,23,26)( 3, 5, 9,10,15,16,19, 24,27)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 7,13,20,25, 6,11,17,22)( 2, 8,14,21,26, 4,12,18,23)( 3, 9,15,19,27, 5,10, 16,24)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 8,15,20,26, 5,11,18,24)( 2, 9,13,21,27, 6,12,16,22)( 3, 7,14,19,25, 4,10, 17,23)$
$ 9, 9, 9 $ $2$ $9$ $( 1, 9,14,20,27, 4,11,16,23)( 2, 7,15,21,25, 5,12,17,24)( 3, 8,13,19,26, 6,10, 18,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,10,21)( 2,11,19)( 3,12,20)( 4,13,24)( 5,14,22)( 6,15,23)( 7,16,26) ( 8,17,27)( 9,18,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,11,20)( 2,12,21)( 3,10,19)( 4,14,23)( 5,15,24)( 6,13,22)( 7,17,25) ( 8,18,26)( 9,16,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,12,19)( 2,10,20)( 3,11,21)( 4,15,22)( 5,13,23)( 6,14,24)( 7,18,27) ( 8,16,25)( 9,17,26)$
$ 9, 9, 9 $ $2$ $9$ $( 1,13,25,11,22, 7,20, 6,17)( 2,14,26,12,23, 8,21, 4,18)( 3,15,27,10,24, 9,19, 5,16)$
$ 9, 9, 9 $ $2$ $9$ $( 1,14,27,11,23, 9,20, 4,16)( 2,15,25,12,24, 7,21, 5,17)( 3,13,26,10,22, 8,19, 6,18)$
$ 9, 9, 9 $ $2$ $9$ $( 1,15,26,11,24, 8,20, 5,18)( 2,13,27,12,22, 9,21, 6,16)( 3,14,25,10,23, 7,19, 4,17)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $54=2 \cdot 3^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  54.3
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B 3C1 3C-1 6A1 6A-1 9A1 9A2 9A4 9B1 9B-1 9B2 9B-2 9B4 9B-4
Size 1 9 1 1 2 2 2 9 9 2 2 2 2 2 2 2 2 2
2 P 1A 1A 3A-1 3A1 3B 3C-1 3C1 3A1 3A-1 9A2 9A4 9A1 9B1 9B-1 9B4 9B-4 9B-2 9B2
3 P 1A 2A 1A 1A 1A 1A 1A 2A 2A 3B 3B 3B 3B 3B 3B 3B 3B 3B
Type
54.3.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
54.3.1b R 1 −1 1 1 1 1 1 −1 −1 1 1 1 1 1 1 1 1 1
54.3.1c1 C 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
54.3.1c2 C 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
54.3.1d1 C 1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
54.3.1d2 C 1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
54.3.2a R 2 0 2 2 2 2 2 0 0 −1 −1 −1 −1 −1 −1 −1 −1 −1
54.3.2b1 C 2 0 2ζ3−1 2ζ3 2 2ζ3−1 2ζ3 0 0 −1 −1 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
54.3.2b2 C 2 0 2ζ3 2ζ3−1 2 2ζ3 2ζ3−1 0 0 −1 −1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
54.3.2c1 R 2 0 2 2 −1 −1 −1 0 0 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−4+ζ94
54.3.2c2 R 2 0 2 2 −1 −1 −1 0 0 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−2+ζ92
54.3.2c3 R 2 0 2 2 −1 −1 −1 0 0 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−1+ζ9
54.3.2d1 C 2 0 2ζ9−3 2ζ93 −1 −ζ9−3 −ζ93 0 0 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ92+ζ94 ζ9−4−ζ9−ζ94 −ζ9−4−ζ92+ζ94 ζ9−4+ζ9 ζ9−2+ζ9−1 ζ9+ζ92
54.3.2d2 C 2 0 2ζ93 2ζ9−3 −1 −ζ93 −ζ9−3 0 0 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4−ζ9−ζ94 ζ92+ζ94 ζ9−4+ζ9 −ζ9−4−ζ92+ζ94 ζ9+ζ92 ζ9−2+ζ9−1
54.3.2d3 C 2 0 2ζ9−3 2ζ93 −1 −ζ9−3 −ζ93 0 0 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ9−1 ζ9+ζ92 ζ9−4−ζ9−ζ94 ζ92+ζ94 ζ9−4+ζ9 −ζ9−4−ζ92+ζ94
54.3.2d4 C 2 0 2ζ93 2ζ9−3 −1 −ζ93 −ζ9−3 0 0 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9+ζ92 ζ9−2+ζ9−1 ζ92+ζ94 ζ9−4−ζ9−ζ94 −ζ9−4−ζ92+ζ94 ζ9−4+ζ9
54.3.2d5 C 2 0 2ζ9−3 2ζ93 −1 −ζ9−3 −ζ93 0 0 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−4+ζ9 −ζ9−4−ζ92+ζ94 ζ9+ζ92 ζ9−2+ζ9−1 ζ92+ζ94 ζ9−4−ζ9−ζ94
54.3.2d6 C 2 0 2ζ93 2ζ9−3 −1 −ζ93 −ζ9−3 0 0 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 −ζ9−4−ζ92+ζ94 ζ9−4+ζ9 ζ9−2+ζ9−1 ζ9+ζ92 ζ9−4−ζ9−ζ94 ζ92+ζ94

magma: CharacterTable(G);