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Properties

Label 27T41
Degree $27$
Order $162$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $\He_3.S_3$

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Show commands: Magma

magma: G := TransitiveGroup(27, 41);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $41$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\He_3.S_3$
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:  (1,10,25,2,11,26,3,12,27)(4,15,19,5,13,20,6,14,21)(7,16,23,8,17,24,9,18,22), (1,20,14,8,24,10)(2,19,15,7,22,12)(3,21,13,9,23,11)(4,26,16)(5,25,17,6,27,18)
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

27T72

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

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:  162.13
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 3B 3C1 3C-1 6A1 6A-1 9A1 9A2 9A4 9B1 9B-1
Size 1 27 2 6 9 9 27 27 6 6 6 18 18
2 P 1A 1A 3A 3B 3C-1 3C1 3C1 3C-1 9A4 9A1 9A2 9B-1 9B1
3 P 1A 2A 1A 1A 1A 1A 2A 2A 3A 3A 3A 3A 3A
Type
162.13.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
162.13.1b R 1 −1 1 1 1 1 −1 −1 1 1 1 1 1
162.13.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3
162.13.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1
162.13.1d1 C 1 −1 1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 ζ3−1 ζ3
162.13.1d2 C 1 −1 1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 ζ3 ζ3−1
162.13.2a R 2 0 2 2 2 2 0 0 −1 −1 −1 −1 −1
162.13.2b1 C 2 0 2 2 2ζ3−1 2ζ3 0 0 −1 −1 −1 −ζ3−1 −ζ3
162.13.2b2 C 2 0 2 2 2ζ3 2ζ3−1 0 0 −1 −1 −1 −ζ3 −ζ3−1
162.13.6a R 6 0 6 −3 0 0 0 0 0 0 0 0 0
162.13.6b1 R 6 0 −3 0 0 0 0 0 2ζ9−4−ζ9+ζ92+ζ94 −ζ9−4+2ζ9−2ζ92+ζ94 −ζ9−4−ζ9+ζ92−2ζ94 0 0
162.13.6b2 R 6 0 −3 0 0 0 0 0 −ζ9−4+2ζ9−2ζ92+ζ94 −ζ9−4−ζ9+ζ92−2ζ94 2ζ9−4−ζ9+ζ92+ζ94 0 0
162.13.6b3 R 6 0 −3 0 0 0 0 0 −ζ9−4−ζ9+ζ92−2ζ94 2ζ9−4−ζ9+ζ92+ζ94 −ζ9−4+2ζ9−2ζ92+ζ94 0 0

magma: CharacterTable(G);