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Properties

Label 27T30
Degree $27$
Order $108$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times D_9$

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Show commands: Magma

magma: G := TransitiveGroup(27, 30);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $30$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times D_9$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,19,10,3,21,12,2,20,11)(4,17,14,25,22,9,5,18,15,26,23,7,6,16,13,27,24,8), (1,26,4)(2,25,5,3,27,6)(7,24,10,18,14,20)(8,23,11,17,15,19)(9,22,12,16,13,21)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$12$:  $D_{6}$ x 2
$18$:  $D_{9}$
$36$:  $S_3^2$, $D_{18}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 2

Degree 9: $D_{9}$, $S_3^2$

Low degree siblings

18T50, 36T86

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

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:  108.16
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 6A 6B 9A1 9A2 9A4 9B1 9B2 9B4 18A1 18A5 18A7
Size 1 3 9 27 2 2 4 6 18 2 2 2 4 4 4 6 6 6
2 P 1A 1A 1A 1A 3A 3B 3C 3A 3B 9A2 9A4 9A1 9B2 9B4 9B1 9A1 9A4 9A2
3 P 1A 2A 2B 2C 1A 1A 1A 2A 2B 3A 3A 3A 3A 3A 3A 6A 6A 6A
Type
108.16.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
108.16.1b R 1 −1 −1 1 1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1
108.16.1c R 1 −1 1 −1 1 1 1 −1 1 1 1 1 1 1 1 −1 −1 −1
108.16.1d R 1 1 −1 −1 1 1 1 1 −1 1 1 1 1 1 1 1 1 1
108.16.2a R 2 0 2 0 2 −1 −1 0 −1 2 2 2 −1 −1 −1 0 0 0
108.16.2b R 2 2 0 0 2 2 2 2 0 −1 −1 −1 −1 −1 −1 −1 −1 −1
108.16.2c R 2 −2 0 0 2 2 2 −2 0 −1 −1 −1 −1 −1 −1 1 1 1
108.16.2d R 2 0 −2 0 2 −1 −1 0 1 2 2 2 −1 −1 −1 0 0 0
108.16.2e1 R 2 2 0 0 −1 2 −1 −1 0 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94
108.16.2e2 R 2 2 0 0 −1 2 −1 −1 0 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92
108.16.2e3 R 2 2 0 0 −1 2 −1 −1 0 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9
108.16.2f1 R 2 −2 0 0 −1 2 −1 1 0 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94
108.16.2f2 R 2 −2 0 0 −1 2 −1 1 0 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92
108.16.2f3 R 2 −2 0 0 −1 2 −1 1 0 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9
108.16.4a R 4 0 0 0 4 −2 −2 0 0 −2 −2 −2 1 1 1 0 0 0
108.16.4b1 R 4 0 0 0 −2 −2 1 0 0 2ζ9−4+2ζ94 2ζ9−1+2ζ9 2ζ9−2+2ζ92 −ζ9−1−ζ9 −ζ9−2−ζ92 −ζ9−4−ζ94 0 0 0
108.16.4b2 R 4 0 0 0 −2 −2 1 0 0 2ζ9−2+2ζ92 2ζ9−4+2ζ94 2ζ9−1+2ζ9 −ζ9−4−ζ94 −ζ9−1−ζ9 −ζ9−2−ζ92 0 0 0
108.16.4b3 R 4 0 0 0 −2 −2 1 0 0 2ζ9−1+2ζ9 2ζ9−2+2ζ92 2ζ9−4+2ζ94 −ζ9−2−ζ92 −ζ9−4−ζ94 −ζ9−1−ζ9 0 0 0

magma: CharacterTable(G);