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Properties

Label 27T49
Degree $27$
Order $162$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $\He_3.C_6$

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Show commands: Magma

magma: G := TransitiveGroup(27, 49);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $49$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\He_3.C_6$
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:  (1,17,12,27,19,7,2,18,10,25,20,8,3,16,11,26,21,9)(4,23,13,6,22,15,5,24,14), (1,16,10)(2,17,11)(3,18,12)(4,21,15)(5,19,13)(6,20,14)(7,27,22)(8,25,23)(9,26,24)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$
$54$:  $C_3^2 : C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 9: $C_3^2 : S_3 $

Low degree siblings

27T40

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $162=2 \cdot 3^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  162.14
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B 3C 3D1 3D-1 6A1 6A-1 9A1 9A-1 9A2 9A-2 9A4 9A-4 18A1 18A-1 18A5 18A-5 18A7 18A-7
Size 1 9 1 1 6 18 18 18 9 9 3 3 3 3 3 3 9 9 9 9 9 9
2 P 1A 1A 3A-1 3A1 3B 3C 3D-1 3D1 3A1 3A-1 9A2 9A-2 9A4 9A-4 9A-1 9A1 9A-1 9A4 9A2 9A-4 9A-2 9A1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 2A 2A 3A1 3A-1 3A-1 3A1 3A1 3A-1 6A-1 6A1 6A-1 6A-1 6A1 6A1
Type
162.14.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
162.14.1b R 1 −1 1 1 1 1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
162.14.1c1 C 1 1 1 1 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
162.14.1c2 C 1 1 1 1 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
162.14.1d1 C 1 −1 1 1 1 1 ζ3−1 ζ3 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
162.14.1d2 C 1 −1 1 1 1 1 ζ3 ζ3−1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
162.14.2a R 2 0 2 2 2 −1 −1 −1 0 0 2 2 2 2 2 2 0 0 0 0 0 0
162.14.2b1 C 2 0 2 2 2 −1 −ζ3 −ζ3−1 0 0 2ζ3−1 2ζ3 2ζ3 2ζ3−1 2ζ3−1 2ζ3 0 0 0 0 0 0
162.14.2b2 C 2 0 2 2 2 −1 −ζ3−1 −ζ3 0 0 2ζ3 2ζ3−1 2ζ3−1 2ζ3 2ζ3 2ζ3−1 0 0 0 0 0 0
162.14.3a1 C 3 1 3ζ9−3 3ζ93 0 0 0 0 ζ93 ζ9−3 ζ9−4−ζ92 ζ9+2ζ94 ζ9−ζ94 −2ζ9−4−ζ92 ζ9−4+2ζ92 −2ζ9−ζ94 ζ94 ζ9−4 ζ92 ζ9−2 ζ9 ζ9−1
162.14.3a2 C 3 1 3ζ93 3ζ9−3 0 0 0 0 ζ9−3 ζ93 ζ9+2ζ94 ζ9−4−ζ92 −2ζ9−4−ζ92 ζ9−ζ94 −2ζ9−ζ94 ζ9−4+2ζ92 ζ9−4 ζ94 ζ9−2 ζ92 ζ9−1 ζ9
162.14.3a3 C 3 1 3ζ9−3 3ζ93 0 0 0 0 ζ93 ζ9−3 ζ9−4+2ζ92 −2ζ9−ζ94 ζ9+2ζ94 ζ9−4−ζ92 −2ζ9−4−ζ92 ζ9−ζ94 ζ9−2 ζ92 ζ9−1 ζ9 ζ94 ζ9−4
162.14.3a4 C 3 1 3ζ93 3ζ9−3 0 0 0 0 ζ9−3 ζ93 −2ζ9−ζ94 ζ9−4+2ζ92 ζ9−4−ζ92 ζ9+2ζ94 ζ9−ζ94 −2ζ9−4−ζ92 ζ92 ζ9−2 ζ9 ζ9−1 ζ9−4 ζ94
162.14.3a5 C 3 1 3ζ9−3 3ζ93 0 0 0 0 ζ93 ζ9−3 −2ζ9−4−ζ92 ζ9−ζ94 −2ζ9−ζ94 ζ9−4+2ζ92 ζ9−4−ζ92 ζ9+2ζ94 ζ9 ζ9−1 ζ9−4 ζ94 ζ9−2 ζ92
162.14.3a6 C 3 1 3ζ93 3ζ9−3 0 0 0 0 ζ9−3 ζ93 ζ9−ζ94 −2ζ9−4−ζ92 ζ9−4+2ζ92 −2ζ9−ζ94 ζ9+2ζ94 ζ9−4−ζ92 ζ9−1 ζ9 ζ94 ζ9−4 ζ92 ζ9−2
162.14.3b1 C 3 −1 3ζ9−3 3ζ93 0 0 0 0 −ζ93 −ζ9−3 ζ9−4−ζ92 ζ9+2ζ94 ζ9−ζ94 −2ζ9−4−ζ92 ζ9−4+2ζ92 −2ζ9−ζ94 −ζ94 −ζ9−4 −ζ92 −ζ9−2 −ζ9 −ζ9−1
162.14.3b2 C 3 −1 3ζ93 3ζ9−3 0 0 0 0 −ζ9−3 −ζ93 ζ9+2ζ94 ζ9−4−ζ92 −2ζ9−4−ζ92 ζ9−ζ94 −2ζ9−ζ94 ζ9−4+2ζ92 −ζ9−4 −ζ94 −ζ9−2 −ζ92 −ζ9−1 −ζ9
162.14.3b3 C 3 −1 3ζ9−3 3ζ93 0 0 0 0 −ζ93 −ζ9−3 ζ9−4+2ζ92 −2ζ9−ζ94 ζ9+2ζ94 ζ9−4−ζ92 −2ζ9−4−ζ92 ζ9−ζ94 −ζ9−2 −ζ92 −ζ9−1 −ζ9 −ζ94 −ζ9−4
162.14.3b4 C 3 −1 3ζ93 3ζ9−3 0 0 0 0 −ζ9−3 −ζ93 −2ζ9−ζ94 ζ9−4+2ζ92 ζ9−4−ζ92 ζ9+2ζ94 ζ9−ζ94 −2ζ9−4−ζ92 −ζ92 −ζ9−2 −ζ9 −ζ9−1 −ζ9−4 −ζ94
162.14.3b5 C 3 −1 3ζ9−3 3ζ93 0 0 0 0 −ζ93 −ζ9−3 −2ζ9−4−ζ92 ζ9−ζ94 −2ζ9−ζ94 ζ9−4+2ζ92 ζ9−4−ζ92 ζ9+2ζ94 −ζ9 −ζ9−1 −ζ9−4 −ζ94 −ζ9−2 −ζ92
162.14.3b6 C 3 −1 3ζ93 3ζ9−3 0 0 0 0 −ζ9−3 −ζ93 ζ9−ζ94 −2ζ9−4−ζ92 ζ9−4+2ζ92 −2ζ9−ζ94 ζ9+2ζ94 ζ9−4−ζ92 −ζ9−1 −ζ9 −ζ94 −ζ9−4 −ζ92 −ζ9−2
162.14.6a R 6 0 6 6 −3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);