Properties

Label 27T108
Degree $27$
Order $243$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_9^2:C_3$

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Show commands: Magma

magma: G := TransitiveGroup(27, 108);
 

Group action invariants

Degree $n$:  $27$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $108$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_9^2:C_3$
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:  (10,13,18,11,14,16,12,15,17)(19,26,24,21,25,23,20,27,22), (1,22,17)(2,23,18)(3,24,16)(4,27,12)(5,25,10)(6,26,11)(7,20,14)(8,21,15)(9,19,13)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$27$:  $C_3^2:C_3$
$81$:  27T23

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 9: $C_3^2:C_3$

Low degree siblings

27T108 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{27}$ $1$ $1$ $0$ $()$
3A1 $3^{9}$ $1$ $3$ $18$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)$
3A-1 $3^{9}$ $1$ $3$ $18$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)$
3B1 $3^{6},1^{9}$ $3$ $3$ $12$ $(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)$
3B-1 $3^{6},1^{9}$ $3$ $3$ $12$ $(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,26,27)$
3C1 $3^{9}$ $27$ $3$ $18$ $( 1,22,17)( 2,23,18)( 3,24,16)( 4,27,12)( 5,25,10)( 6,26,11)( 7,20,14)( 8,21,15)( 9,19,13)$
3C-1 $3^{9}$ $27$ $3$ $18$ $( 1,17,22)( 2,18,23)( 3,16,24)( 4,12,27)( 5,10,25)( 6,11,26)( 7,14,20)( 8,15,21)( 9,13,19)$
9A1 $9^{3}$ $3$ $9$ $24$ $( 1, 9, 6, 3, 8, 5, 2, 7, 4)(10,18,14,12,17,13,11,16,15)(19,27,23,21,26,22,20,25,24)$
9A-1 $9^{3}$ $3$ $9$ $24$ $( 1, 7, 5, 3, 9, 4, 2, 8, 6)(10,16,13,12,18,15,11,17,14)(19,25,22,21,27,24,20,26,23)$
9A2 $9^{2},3^{3}$ $3$ $9$ $22$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,18,14,12,17,13,11,16,15)(19,23,26,20,24,27,21,22,25)$
9A-2 $9^{2},3^{3}$ $3$ $9$ $22$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,18,14,12,17,13,11,16,15)(19,22,27,20,23,25,21,24,26)$
9A4 $9^{3}$ $3$ $9$ $24$ $( 1, 5, 9, 2, 6, 7, 3, 4, 8)(10,13,18,11,14,16,12,15,17)(19,22,27,20,23,25,21,24,26)$
9A-4 $9^{2},3^{3}$ $3$ $9$ $22$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,14,17,11,15,18,12,13,16)(19,26,24,21,25,23,20,27,22)$
9B1 $9^{2},3^{3}$ $3$ $9$ $22$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,13,18,11,14,16,12,15,17)(19,25,22,21,27,24,20,26,23)$
9B-1 $9^{2},3^{3}$ $3$ $9$ $22$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,17,15,12,16,14,11,18,13)(19,24,25,20,22,26,21,23,27)$
9B2 $9^{2},3^{3}$ $3$ $9$ $22$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,16,13,12,18,15,11,17,14)(19,24,25,20,22,26,21,23,27)$
9B-2 $9^{2},3^{3}$ $3$ $9$ $22$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,17,15,12,16,14,11,18,13)(19,23,26,20,24,27,21,22,25)$
9B4 $9^{2},1^{9}$ $3$ $9$ $16$ $(10,13,18,11,14,16,12,15,17)(19,26,24,21,25,23,20,27,22)$
9B-4 $9^{2},1^{9}$ $3$ $9$ $16$ $(10,14,17,11,15,18,12,13,16)(19,25,22,21,27,24,20,26,23)$
9C1 $9^{2},3^{3}$ $3$ $9$ $22$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,15,16,11,13,17,12,14,18)(19,26,24,21,25,23,20,27,22)$
9C-1 $9^{3}$ $3$ $9$ $24$ $( 1, 8, 4, 3, 7, 6, 2, 9, 5)(10,17,15,12,16,14,11,18,13)(19,26,24,21,25,23,20,27,22)$
9C2 $9^{2},3^{3}$ $3$ $9$ $22$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,13,18,11,14,16,12,15,17)(19,27,23,21,26,22,20,25,24)$
9C-2 $9^{3}$ $3$ $9$ $24$ $( 1, 4, 7, 2, 5, 8, 3, 6, 9)(10,15,16,11,13,17,12,14,18)(19,24,25,20,22,26,21,23,27)$
9C4 $9^{2},1^{9}$ $3$ $9$ $16$ $(10,16,13,12,18,15,11,17,14)(19,23,26,20,24,27,21,22,25)$
9C-4 $9^{2},3^{3}$ $3$ $9$ $22$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,14,17,11,15,18,12,13,16)(19,27,23,21,26,22,20,25,24)$
9D1 $9^{2},1^{9}$ $3$ $9$ $16$ $(10,15,16,11,13,17,12,14,18)(19,27,23,21,26,22,20,25,24)$
9D-1 $9^{2},1^{9}$ $3$ $9$ $16$ $(10,18,14,12,17,13,11,16,15)(19,24,25,20,22,26,21,23,27)$
9D2 $9^{2},3^{3}$ $3$ $9$ $22$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,15,16,11,13,17,12,14,18)(19,25,22,21,27,24,20,26,23)$
9D-2 $9^{2},1^{9}$ $3$ $9$ $16$ $(10,17,15,12,16,14,11,18,13)(19,22,27,20,23,25,21,24,26)$
9D4 $9^{2},3^{3}$ $3$ $9$ $22$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,16,13,12,18,15,11,17,14)(19,22,27,20,23,25,21,24,26)$
9D-4 $9^{3}$ $3$ $9$ $24$ $( 1, 6, 8, 2, 4, 9, 3, 5, 7)(10,14,17,11,15,18,12,13,16)(19,23,26,20,24,27,21,22,25)$
9E1 $9^{3}$ $27$ $9$ $24$ $( 1,26,16, 3,25,18, 2,27,17)( 4,19,11, 6,21,10, 5,20,12)( 7,22,13, 9,24,15, 8,23,14)$
9E-1 $9^{3}$ $27$ $9$ $24$ $( 1,10,23, 2,11,24, 3,12,22)( 4,15,25, 5,13,26, 6,14,27)( 7,16,21, 8,17,19, 9,18,20)$
9F1 $9^{3}$ $27$ $9$ $24$ $( 1,19,18, 2,20,16, 3,21,17)( 4,22,10, 5,23,11, 6,24,12)( 7,27,15, 8,25,13, 9,26,14)$
9F-1 $9^{3}$ $27$ $9$ $24$ $( 1,13,24, 3,15,23, 2,14,22)( 4,17,26, 6,16,25, 5,18,27)( 7,12,19, 9,11,21, 8,10,20)$

Malle's constant $a(G)$:     $1/12$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $243=3^{5}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $4$
Label:  243.25
magma: IdentifyGroup(G);
 
Character table:    35 x 35 character table

magma: CharacterTable(G);