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Properties

Label 27T9
27T9 1 2 1->2 22 1->22 3 2->3 23 2->23 3->1 24 3->24 4 26 4->26 27 4->27 5 25 5->25 5->27 6 6->25 6->26 7 7->1 7->23 8 8->2 8->24 9 9->3 9->22 10 10->5 20 10->20 11 11->6 21 11->21 12 12->4 19 12->19 13 13->7 18 13->18 14 14->8 16 14->16 15 15->9 17 15->17 16->10 16->13 17->11 17->14 18->12 18->15 19->11 19->15 20->12 20->13 21->10 21->14 22->8 22->17 23->9 23->18 24->7 24->16 25->4 25->20 26->5 26->21 27->6 27->19
Degree $27$
Order $54$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times D_9$

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Copy content magma:G := TransitiveGroup(27, 9);
 

Group invariants

Abstract group:  $C_3\times D_9$
Copy content magma:IdentifyGroup(G);
 
Order:  $54=2 \cdot 3^{3}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$, $D_{9}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$, $S_3$

Degree 9: $D_{9}$, $S_3\times C_3$

Low degree siblings

18T19

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 3B 3C1 3C-1 6A1 6A-1 9A1 9A2 9A4 9B1 9B-1 9B2 9B-2 9B4 9B-4
Size 1 9 1 1 2 2 2 9 9 2 2 2 2 2 2 2 2 2
2 P 1A 1A 3A-1 3A1 3B 3C-1 3C1 3A1 3A-1 9A2 9A4 9A1 9B2 9B-2 9B4 9B-4 9B-1 9B1
3 P 1A 2A 1A 1A 1A 1A 1A 2A 2A 3B 3B 3B 3B 3B 3B 3B 3B 3B
Type
54.3.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
54.3.1b R 1 −1 1 1 1 1 1 −1 −1 1 1 1 1 1 1 1 1 1
54.3.1c1 C 1 1 ζ3−1 ζ3 ζ3−1 1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3
54.3.1c2 C 1 1 ζ3 ζ3−1 ζ3 1 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1
54.3.1d1 C 1 −1 ζ3−1 ζ3 ζ3−1 1 ζ3 −ζ3 −ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3
54.3.1d2 C 1 −1 ζ3 ζ3−1 ζ3 1 ζ3−1 −ζ3−1 −ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1
54.3.2a R 2 0 2 2 2 2 2 0 0 −1 −1 −1 −1 −1 −1 −1 −1 −1
54.3.2b1 C 2 0 2ζ3−1 2ζ3 2ζ3−1 2 2ζ3 0 0 −1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3
54.3.2b2 C 2 0 2ζ3 2ζ3−1 2ζ3 2 2ζ3−1 0 0 −1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1
54.3.2c1 R 2 0 2 2 −1 −1 −1 0 0 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−1+ζ9
54.3.2c2 R 2 0 2 2 −1 −1 −1 0 0 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−4+ζ94
54.3.2c3 R 2 0 2 2 −1 −1 −1 0 0 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−2+ζ92
54.3.2d1 C 2 0 2ζ9−3 2ζ93 −ζ9−3 −1 −ζ93 0 0 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−4+ζ9 ζ9+ζ92 ζ9−2+ζ9−1 −ζ9−4−ζ92+ζ94 ζ9−1+ζ9 ζ9−4−ζ9−ζ94 ζ92+ζ94
54.3.2d2 C 2 0 2ζ93 2ζ9−3 −ζ93 −1 −ζ9−3 0 0 ζ9−4+ζ94 ζ9−2+ζ92 −ζ9−4−ζ92+ζ94 ζ9−2+ζ9−1 ζ9+ζ92 ζ9−4+ζ9 ζ9−1+ζ9 ζ92+ζ94 ζ9−4−ζ9−ζ94
54.3.2d3 C 2 0 2ζ9−3 2ζ93 −ζ9−3 −1 −ζ93 0 0 ζ9−2+ζ92 ζ9−1+ζ9 ζ92+ζ94 −ζ9−4−ζ92+ζ94 ζ9−4+ζ9 ζ9−4−ζ9−ζ94 ζ9−4+ζ94 ζ9+ζ92 ζ9−2+ζ9−1
54.3.2d4 C 2 0 2ζ93 2ζ9−3 −ζ93 −1 −ζ9−3 0 0 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4−ζ9−ζ94 ζ9−4+ζ9 −ζ9−4−ζ92+ζ94 ζ92+ζ94 ζ9−4+ζ94 ζ9−2+ζ9−1 ζ9+ζ92
54.3.2d5 C 2 0 2ζ9−3 2ζ93 −ζ9−3 −1 −ζ93 0 0 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ9−1 ζ9−4−ζ9−ζ94 ζ92+ζ94 ζ9+ζ92 ζ9−2+ζ92 −ζ9−4−ζ92+ζ94 ζ9−4+ζ9
54.3.2d6 C 2 0 2ζ93 2ζ9−3 −ζ93 −1 −ζ9−3 0 0 ζ9−1+ζ9 ζ9−4+ζ94 ζ9+ζ92 ζ92+ζ94 ζ9−4−ζ9−ζ94 ζ9−2+ζ9−1 ζ9−2+ζ92 ζ9−4+ζ9 −ζ9−4−ζ92+ζ94

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed