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Properties

Label 27T40
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, 40);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $40$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\He_3.C_6$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $9$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,23,15)(2,24,13)(3,22,14)(4,26,17)(5,27,18)(6,25,16)(7,21,12)(8,19,10)(9,20,11), (1,25)(2,26)(3,27)(4,21)(5,19)(6,20)(7,22)(8,23)(9,24)
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: $C_3$, $S_3$

Degree 9: $S_3\times C_3$

Low degree siblings

27T49

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

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);