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Properties

Label 27T45
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, 45);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $45$
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:  (1,5,7,2,6,8,3,4,9)(10,21,16,27,13,23,12,20,18,26,15,22,11,19,17,25,14,24), (1,18,21,2,16,19,3,17,20)(4,10,22,5,11,23,6,12,24)(7,14,26,8,15,27,9,13,25)
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

27T69

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:  162.12
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B 3C 6A1 6A-1 9A1 9A-1 9A2 9A-2 9A4 9A-4 9B1 9B-1 18A1 18A-1 18A5 18A-5 18A7 18A-7
Size 1 9 1 1 6 18 9 9 3 3 3 3 3 3 18 18 9 9 9 9 9 9
2 P 1A 1A 3A-1 3A1 3B 3C 3A1 3A-1 9A-1 9A-2 9A2 9A-4 9A4 9A1 9B-1 9B1 9A-4 9A2 9A1 9A4 9A-2 9A-1
3 P 1A 2A 1A 1A 1A 1A 2A 2A 3A1 3A-1 3A1 3A1 3A-1 3A-1 3A1 3A-1 6A-1 6A-1 6A1 6A1 6A1 6A-1
Type
162.12.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
162.12.1b R 1 −1 1 1 1 1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
162.12.1c1 C 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
162.12.1c2 C 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
162.12.1d1 C 1 −1 1 1 1 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
162.12.1d2 C 1 −1 1 1 1 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
162.12.2a R 2 0 2 2 2 −1 0 0 2 2 2 2 2 2 −1 −1 0 0 0 0 0 0
162.12.2b1 C 2 0 2 2 2 −1 0 0 2ζ3−1 2ζ3 2ζ3 2ζ3−1 2ζ3−1 2ζ3 −ζ3 −ζ3−1 0 0 0 0 0 0
162.12.2b2 C 2 0 2 2 2 −1 0 0 2ζ3 2ζ3−1 2ζ3−1 2ζ3 2ζ3 2ζ3−1 −ζ3−1 −ζ3 0 0 0 0 0 0
162.12.3a1 C 3 1 3ζ9−3 3ζ93 0 0 ζ93 ζ9−3 2ζ9−4+ζ92 −ζ9+ζ94 2ζ9+ζ94 −ζ9−4−2ζ92 −ζ9−4+ζ92 −ζ9−2ζ94 0 0 ζ9 ζ9−1 ζ9−4 ζ94 ζ9−2 ζ92
162.12.3a2 C 3 1 3ζ93 3ζ9−3 0 0 ζ9−3 ζ93 −ζ9+ζ94 2ζ9−4+ζ92 −ζ9−4−2ζ92 2ζ9+ζ94 −ζ9−2ζ94 −ζ9−4+ζ92 0 0 ζ9−1 ζ9 ζ94 ζ9−4 ζ92 ζ9−2
162.12.3a3 C 3 1 3ζ9−3 3ζ93 0 0 ζ93 ζ9−3 −ζ9−4−2ζ92 2ζ9+ζ94 −ζ9−2ζ94 −ζ9−4+ζ92 2ζ9−4+ζ92 −ζ9+ζ94 0 0 ζ9−2 ζ92 ζ9−1 ζ9 ζ94 ζ9−4
162.12.3a4 C 3 1 3ζ93 3ζ9−3 0 0 ζ9−3 ζ93 2ζ9+ζ94 −ζ9−4−2ζ92 −ζ9−4+ζ92 −ζ9−2ζ94 −ζ9+ζ94 2ζ9−4+ζ92 0 0 ζ92 ζ9−2 ζ9 ζ9−1 ζ9−4 ζ94
162.12.3a5 C 3 1 3ζ9−3 3ζ93 0 0 ζ93 ζ9−3 −ζ9−4+ζ92 −ζ9−2ζ94 −ζ9+ζ94 2ζ9−4+ζ92 −ζ9−4−2ζ92 2ζ9+ζ94 0 0 ζ94 ζ9−4 ζ92 ζ9−2 ζ9 ζ9−1
162.12.3a6 C 3 1 3ζ93 3ζ9−3 0 0 ζ9−3 ζ93 −ζ9−2ζ94 −ζ9−4+ζ92 2ζ9−4+ζ92 −ζ9+ζ94 2ζ9+ζ94 −ζ9−4−2ζ92 0 0 ζ9−4 ζ94 ζ9−2 ζ92 ζ9−1 ζ9
162.12.3b1 C 3 −1 3ζ9−3 3ζ93 0 0 −ζ93 −ζ9−3 2ζ9−4+ζ92 −ζ9+ζ94 2ζ9+ζ94 −ζ9−4−2ζ92 −ζ9−4+ζ92 −ζ9−2ζ94 0 0 −ζ9 −ζ9−1 −ζ9−4 −ζ94 −ζ9−2 −ζ92
162.12.3b2 C 3 −1 3ζ93 3ζ9−3 0 0 −ζ9−3 −ζ93 −ζ9+ζ94 2ζ9−4+ζ92 −ζ9−4−2ζ92 2ζ9+ζ94 −ζ9−2ζ94 −ζ9−4+ζ92 0 0 −ζ9−1 −ζ9 −ζ94 −ζ9−4 −ζ92 −ζ9−2
162.12.3b3 C 3 −1 3ζ9−3 3ζ93 0 0 −ζ93 −ζ9−3 −ζ9−4−2ζ92 2ζ9+ζ94 −ζ9−2ζ94 −ζ9−4+ζ92 2ζ9−4+ζ92 −ζ9+ζ94 0 0 −ζ9−2 −ζ92 −ζ9−1 −ζ9 −ζ94 −ζ9−4
162.12.3b4 C 3 −1 3ζ93 3ζ9−3 0 0 −ζ9−3 −ζ93 2ζ9+ζ94 −ζ9−4−2ζ92 −ζ9−4+ζ92 −ζ9−2ζ94 −ζ9+ζ94 2ζ9−4+ζ92 0 0 −ζ92 −ζ9−2 −ζ9 −ζ9−1 −ζ9−4 −ζ94
162.12.3b5 C 3 −1 3ζ9−3 3ζ93 0 0 −ζ93 −ζ9−3 −ζ9−4+ζ92 −ζ9−2ζ94 −ζ9+ζ94 2ζ9−4+ζ92 −ζ9−4−2ζ92 2ζ9+ζ94 0 0 −ζ94 −ζ9−4 −ζ92 −ζ9−2 −ζ9 −ζ9−1
162.12.3b6 C 3 −1 3ζ93 3ζ9−3 0 0 −ζ9−3 −ζ93 −ζ9−2ζ94 −ζ9−4+ζ92 2ζ9−4+ζ92 −ζ9+ζ94 2ζ9+ζ94 −ζ9−4−2ζ92 0 0 −ζ9−4 −ζ94 −ζ9−2 −ζ92 −ζ9−1 −ζ9
162.12.6a 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);