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Properties

Label 27T50
27T50 1 3 1->3 11 1->11 2 10 2->10 12 3->12 4 6 4->6 14 4->14 5 13 5->13 15 6->15 7 16 7->16 27 7->27 8 18 8->18 26 8->26 9 17 9->17 25 9->25 10->3 19 10->19 11->2 21 11->21 12->1 20 12->20 13->6 22 13->22 14->5 24 14->24 15->4 23 15->23 16->15 16->20 17->14 17->19 18->13 18->21 19->9 19->23 20->8 20->22 21->7 21->24 22->12 22->17 23->11 23->16 24->10 24->18 25->8 25->27 26->7 27->9
Degree $27$
Order $162$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\wr S_3$

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Copy content magma:G := TransitiveGroup(27, 50);
 

Group invariants

Abstract group:  $C_3\wr S_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $162=2 \cdot 3^{4}$
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 $t$:  $50$
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:  $(1,11,2,10,3,12)(4,14,5,13,6,15)(7,27,9,25,8,26)(16,20,22,17,19,23)(18,21,24)$, $(1,3)(4,6)(7,16,15,23,11,21)(8,18,13,22,12,20)(9,17,14,24,10,19)(25,27)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$
$54$:  $C_3^2 : C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 9: $C_3^2 : C_6$, $C_3 \wr S_3 $

Low degree siblings

9T20 x 3, 18T86 x 3, 27T37, 27T50 x 2, 27T70

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 $2^{12},1^{3}$ $9$ $2$ $12$ $( 1,14)( 2,13)( 3,15)( 4, 8)( 5, 7)( 6, 9)(10,26)(11,25)(12,27)(16,18)(19,21)(22,24)$
3A1 $3^{9}$ $1$ $3$ $18$ $( 1,25, 4)( 2,26, 5)( 3,27, 6)( 7,13,10)( 8,14,11)( 9,15,12)(16,22,19)(17,23,20)(18,24,21)$
3A-1 $3^{9}$ $1$ $3$ $18$ $( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7,10,13)( 8,11,14)( 9,12,15)(16,19,22)(17,20,23)(18,21,24)$
3B1 $3^{9}$ $3$ $3$ $18$ $( 1, 6,26)( 2, 4,27)( 3, 5,25)( 7,11,15)( 8,12,13)( 9,10,14)(16,22,19)(17,23,20)(18,24,21)$
3B-1 $3^{9}$ $3$ $3$ $18$ $( 1,26, 6)( 2,27, 4)( 3,25, 5)( 7,15,11)( 8,13,12)( 9,14,10)(16,19,22)(17,20,23)(18,21,24)$
3C1 $3^{9}$ $3$ $3$ $18$ $( 1, 2, 3)( 4, 5, 6)( 7, 9, 8)(10,12,11)(13,15,14)(16,22,19)(17,23,20)(18,24,21)(25,26,27)$
3C-1 $3^{9}$ $3$ $3$ $18$ $( 1, 3, 2)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,14,15)(16,19,22)(17,20,23)(18,21,24)(25,27,26)$
3D1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 1,27, 5)( 2,25, 6)( 3,26, 4)( 7,14,12)( 8,15,10)( 9,13,11)$
3D-1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 1, 5,27)( 2, 6,25)( 3, 4,26)( 7,12,14)( 8,10,15)( 9,11,13)$
3E $3^{9}$ $6$ $3$ $18$ $( 1,27, 5)( 2,25, 6)( 3,26, 4)( 7, 9, 8)(10,12,11)(13,15,14)(16,21,23)(17,19,24)(18,20,22)$
3F $3^{9}$ $18$ $3$ $18$ $( 1,17,11)( 2,18,12)( 3,16,10)( 4,20,14)( 5,21,15)( 6,19,13)( 7,27,22)( 8,25,23)( 9,26,24)$
6A1 $6^{4},3$ $9$ $6$ $22$ $( 1, 8,25,14, 4,11)( 2, 7,26,13, 5,10)( 3, 9,27,15, 6,12)(16,21,22,18,19,24)(17,20,23)$
6A-1 $6^{4},3$ $9$ $6$ $22$ $( 1,11, 4,14,25, 8)( 2,10, 5,13,26, 7)( 3,12, 6,15,27, 9)(16,24,19,18,22,21)(17,23,20)$
6B1 $6^{4},3$ $9$ $6$ $22$ $( 1,10, 6,14,26, 9)( 2,12, 4,13,27, 8)( 3,11, 5,15,25, 7)(16,21,22,18,19,24)(17,20,23)$
6B-1 $6^{4},3$ $9$ $6$ $22$ $( 1, 9,26,14, 6,10)( 2, 8,27,13, 4,12)( 3, 7,25,15, 5,11)(16,24,19,18,22,21)(17,23,20)$
6C1 $6^{4},3$ $9$ $6$ $22$ $( 1,15, 2,14, 3,13)( 4, 9, 5, 8, 6, 7)(10,25,12,26,11,27)(16,21,22,18,19,24)(17,20,23)$
6C-1 $6^{4},3$ $9$ $6$ $22$ $( 1,13, 3,14, 2,15)( 4, 7, 6, 8, 5, 9)(10,27,11,26,12,25)(16,24,19,18,22,21)(17,23,20)$
6D1 $6^{3},2^{3},1^{3}$ $9$ $6$ $18$ $( 1, 7,27,14, 5,12)( 2, 9,25,13, 6,11)( 3, 8,26,15, 4,10)(16,18)(19,21)(22,24)$
6D-1 $6^{3},2^{3},1^{3}$ $9$ $6$ $18$ $( 1,12, 5,14,27, 7)( 2,11, 6,13,25, 9)( 3,10, 4,15,26, 8)(16,18)(19,21)(22,24)$
9A1 $9^{3}$ $18$ $9$ $24$ $( 1, 8,16, 4,11,19,25,14,22)( 2, 9,17, 5,12,20,26,15,23)( 3, 7,18, 6,10,21,27,13,24)$
9A-1 $9^{3}$ $18$ $9$ $24$ $( 1,22,14,25,19,11, 4,16, 8)( 2,23,15,26,20,12, 5,17, 9)( 3,24,13,27,21,10, 6,18, 7)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 3E 3F 6A1 6A-1 6B1 6B-1 6C1 6C-1 6D1 6D-1 9A1 9A-1
Size 1 9 1 1 3 3 3 3 3 3 6 18 9 9 9 9 9 9 9 9 18 18
2 P 1A 1A 3A-1 3A1 3B-1 3B1 3C-1 3C1 3D-1 3D1 3E 3F 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 9A-1 9A1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2A 2A 3A-1 3A1
Type
162.10.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
162.10.1b R 1 −1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1
162.10.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3−1 ζ3 ζ3−1
162.10.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3
162.10.1d1 C 1 −1 1 1 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 1 1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −1 −ζ3−1 ζ3 ζ3−1
162.10.1d2 C 1 −1 1 1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 1 1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −1 −ζ3 ζ3−1 ζ3
162.10.2a R 2 0 2 2 2 2 2 2 2 2 2 −1 0 0 0 0 0 0 0 0 −1 −1
162.10.2b1 C 2 0 2 2 2ζ3−1 2ζ3 2ζ3−1 2ζ3−1 2ζ3 2ζ3 2 −1 0 0 0 0 0 0 0 0 −ζ3 −ζ3−1
162.10.2b2 C 2 0 2 2 2ζ3 2ζ3−1 2ζ3 2ζ3 2ζ3−1 2ζ3−1 2 −1 0 0 0 0 0 0 0 0 −ζ3−1 −ζ3
162.10.3a1 C 3 1 3ζ3−1 3ζ3 −2−ζ3 −1+ζ3 1−ζ3 1+2ζ3 −1−2ζ3 2+ζ3 0 0 ζ3 ζ3 ζ3 1 1 ζ3−1 ζ3−1 ζ3−1 0 0
162.10.3a2 C 3 1 3ζ3 3ζ3−1 −1+ζ3 −2−ζ3 2+ζ3 −1−2ζ3 1+2ζ3 1−ζ3 0 0 ζ3−1 ζ3−1 ζ3−1 1 1 ζ3 ζ3 ζ3 0 0
162.10.3b1 C 3 1 3ζ3−1 3ζ3 1+2ζ3 −1−2ζ3 −2−ζ3 1−ζ3 2+ζ3 −1+ζ3 0 0 ζ3 ζ3−1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 0 0
162.10.3b2 C 3 1 3ζ3 3ζ3−1 −1−2ζ3 1+2ζ3 −1+ζ3 2+ζ3 1−ζ3 −2−ζ3 0 0 ζ3−1 ζ3 1 ζ3−1 ζ3 1 ζ3 ζ3−1 0 0
162.10.3c1 C 3 1 3ζ3−1 3ζ3 1−ζ3 2+ζ3 1+2ζ3 −2−ζ3 −1+ζ3 −1−2ζ3 0 0 ζ3 1 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 0 0
162.10.3c2 C 3 1 3ζ3 3ζ3−1 2+ζ3 1−ζ3 −1−2ζ3 −1+ζ3 −2−ζ3 1+2ζ3 0 0 ζ3−1 1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 0 0
162.10.3d1 C 3 −1 3ζ3−1 3ζ3 −2−ζ3 −1+ζ3 1−ζ3 1+2ζ3 −1−2ζ3 2+ζ3 0 0 −ζ3 −ζ3 −ζ3 −1 −1 −ζ3−1 −ζ3−1 −ζ3−1 0 0
162.10.3d2 C 3 −1 3ζ3 3ζ3−1 −1+ζ3 −2−ζ3 2+ζ3 −1−2ζ3 1+2ζ3 1−ζ3 0 0 −ζ3−1 −ζ3−1 −ζ3−1 −1 −1 −ζ3 −ζ3 −ζ3 0 0
162.10.3e1 C 3 −1 3ζ3−1 3ζ3 1+2ζ3 −1−2ζ3 −2−ζ3 1−ζ3 2+ζ3 −1+ζ3 0 0 −ζ3 −ζ3−1 −1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3 0 0
162.10.3e2 C 3 −1 3ζ3 3ζ3−1 −1−2ζ3 1+2ζ3 −1+ζ3 2+ζ3 1−ζ3 −2−ζ3 0 0 −ζ3−1 −ζ3 −1 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1 0 0
162.10.3f1 C 3 −1 3ζ3−1 3ζ3 1−ζ3 2+ζ3 1+2ζ3 −2−ζ3 −1+ζ3 −1−2ζ3 0 0 −ζ3 −1 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −1 0 0
162.10.3f2 C 3 −1 3ζ3 3ζ3−1 2+ζ3 1−ζ3 −1−2ζ3 −1+ζ3 −2−ζ3 1+2ζ3 0 0 −ζ3−1 −1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −1 0 0
162.10.6a R 6 0 6 6 0 0 0 0 0 0 −3 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed