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Properties

Label 27T12
Degree $27$
Order $54$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times C_9$

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Show commands: Magma

magma: G := TransitiveGroup(27, 12);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $12$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times C_9$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $9$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,11,20,2,12,21,3,10,19)(4,9,23,25,13,17,6,8,22,27,15,16,5,7,24,26,14,18), (1,26,6)(2,27,4)(3,25,5)(7,13,10)(8,14,11)(9,15,12)(16,23,21)(17,24,19)(18,22,20)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$9$:  $C_9$
$18$:  $S_3\times C_3$, $C_{18}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$, $S_3$

Degree 9: $C_9$, $S_3\times C_3$

Low degree siblings

18T16

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

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:  54.4
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B 3C1 3C-1 6A1 6A-1 9A1 9A-1 9A2 9A-2 9A4 9A-4 9B1 9B-1 9B2 9B-2 9B4 9B-4 18A1 18A-1 18A5 18A-5 18A7 18A-7
Size 1 3 1 1 2 2 2 3 3 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3
2 P 1A 1A 3A-1 3A1 3C1 3B 3C-1 3A1 3A-1 9A2 9A-2 9A1 9A-4 9A4 9A-1 9B1 9B-2 9B4 9B-4 9B2 9B-1 9A2 9A1 9A-4 9A4 9A-2 9A-1
3 P 1A 2A 1A 1A 1A 1A 1A 2A 2A 3A1 3A-1 3A-1 3A1 3A-1 3A1 3A-1 3A-1 3A-1 3A1 3A1 3A1 6A-1 6A1 6A-1 6A1 6A1 6A-1
Type
54.4.1a R 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
54.4.1b R 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
54.4.1c1 C 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
54.4.1c2 C 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
54.4.1d1 C 1 −1 1 1 1 1 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
54.4.1d2 C 1 −1 1 1 1 1 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
54.4.1e1 C 1 1 ζ9−3 ζ93 1 ζ93 ζ9−3 ζ93 ζ9−3 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−1 ζ9 ζ94 ζ9−4
54.4.1e2 C 1 1 ζ93 ζ9−3 1 ζ9−3 ζ93 ζ9−3 ζ93 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2 ζ9 ζ9−1 ζ9−4 ζ94
54.4.1e3 C 1 1 ζ9−3 ζ93 1 ζ93 ζ9−3 ζ93 ζ9−3 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1 ζ9−4 ζ94 ζ9−2 ζ92
54.4.1e4 C 1 1 ζ93 ζ9−3 1 ζ9−3 ζ93 ζ9−3 ζ93 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9 ζ94 ζ9−4 ζ92 ζ9−2
54.4.1e5 C 1 1 ζ9−3 ζ93 1 ζ93 ζ9−3 ζ93 ζ9−3 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4 ζ92 ζ9−2 ζ9 ζ9−1
54.4.1e6 C 1 1 ζ93 ζ9−3 1 ζ9−3 ζ93 ζ9−3 ζ93 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94 ζ9−2 ζ92 ζ9−1 ζ9
54.4.1f1 C 1 −1 ζ9−3 ζ93 1 ζ93 ζ9−3 −ζ93 −ζ9−3 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9 −ζ9−2 −ζ92 −ζ9−1 −ζ9 −ζ94 −ζ9−4
54.4.1f2 C 1 −1 ζ93 ζ9−3 1 ζ9−3 ζ93 −ζ9−3 −ζ93 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1 −ζ92 −ζ9−2 −ζ9 −ζ9−1 −ζ9−4 −ζ94
54.4.1f3 C 1 −1 ζ9−3 ζ93 1 ζ93 ζ9−3 −ζ93 −ζ9−3 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94 −ζ9 −ζ9−1 −ζ9−4 −ζ94 −ζ9−2 −ζ92
54.4.1f4 C 1 −1 ζ93 ζ9−3 1 ζ9−3 ζ93 −ζ9−3 −ζ93 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4 −ζ9−1 −ζ9 −ζ94 −ζ9−4 −ζ92 −ζ9−2
54.4.1f5 C 1 −1 ζ9−3 ζ93 1 ζ93 ζ9−3 −ζ93 −ζ9−3 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2 −ζ94 −ζ9−4 −ζ92 −ζ9−2 −ζ9 −ζ9−1
54.4.1f6 C 1 −1 ζ93 ζ9−3 1 ζ9−3 ζ93 −ζ9−3 −ζ93 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92 −ζ9−4 −ζ94 −ζ9−2 −ζ92 −ζ9−1 −ζ9
54.4.2a R 2 0 2 2 −1 −1 −1 0 0 2 2 2 2 2 2 −1 −1 −1 −1 −1 −1 0 0 0 0 0 0
54.4.2b1 C 2 0 2 2 −1 −1 −1 0 0 2ζ3−1 2ζ3 2ζ3 2ζ3−1 2ζ3−1 2ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 0 0 0 0 0 0
54.4.2b2 C 2 0 2 2 −1 −1 −1 0 0 2ζ3 2ζ3−1 2ζ3−1 2ζ3 2ζ3 2ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 0 0 0 0 0 0
54.4.2c1 C 2 0 2ζ9−3 2ζ93 −1 −ζ93 −ζ9−3 0 0 2ζ9−4 2ζ94 2ζ9 2ζ9−1 2ζ92 2ζ9−2 −ζ92 −ζ9−2 −ζ94 −ζ9−4 −ζ9−1 −ζ9 0 0 0 0 0 0
54.4.2c2 C 2 0 2ζ93 2ζ9−3 −1 −ζ9−3 −ζ93 0 0 2ζ94 2ζ9−4 2ζ9−1 2ζ9 2ζ9−2 2ζ92 −ζ9−2 −ζ92 −ζ9−4 −ζ94 −ζ9 −ζ9−1 0 0 0 0 0 0
54.4.2c3 C 2 0 2ζ9−3 2ζ93 −1 −ζ93 −ζ9−3 0 0 2ζ92 2ζ9−2 2ζ94 2ζ9−4 2ζ9−1 2ζ9 −ζ9−1 −ζ9 −ζ9−2 −ζ92 −ζ9−4 −ζ94 0 0 0 0 0 0
54.4.2c4 C 2 0 2ζ93 2ζ9−3 −1 −ζ9−3 −ζ93 0 0 2ζ9−2 2ζ92 2ζ9−4 2ζ94 2ζ9 2ζ9−1 −ζ9 −ζ9−1 −ζ92 −ζ9−2 −ζ94 −ζ9−4 0 0 0 0 0 0
54.4.2c5 C 2 0 2ζ9−3 2ζ93 −1 −ζ93 −ζ9−3 0 0 2ζ9−1 2ζ9 2ζ9−2 2ζ92 2ζ9−4 2ζ94 −ζ9−4 −ζ94 −ζ9 −ζ9−1 −ζ92 −ζ9−2 0 0 0 0 0 0
54.4.2c6 C 2 0 2ζ93 2ζ9−3 −1 −ζ9−3 −ζ93 0 0 2ζ9 2ζ9−1 2ζ92 2ζ9−2 2ζ94 2ζ9−4 −ζ94 −ζ9−4 −ζ9−1 −ζ9 −ζ9−2 −ζ92 0 0 0 0 0 0

magma: CharacterTable(G);