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Properties

Label 27T47
27T47 1 8 1->8 2 9 2->9 3 7 3->7 4 15 4->15 5 13 5->13 6 14 6->14 10 7->10 24 7->24 11 8->11 22 8->22 12 9->12 23 9->23 10->13 19 10->19 11->14 20 11->20 12->15 21 12->21 13->7 17 13->17 14->8 18 14->18 15->9 16 15->16 16->23 25 16->25 17->24 26 17->26 18->22 27 18->27 19->6 19->17 20->4 20->18 21->5 21->16 22->2 22->20 23->3 23->21 24->1 24->19 25->10 26->11 27->12
Degree $27$
Order $162$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_3^2:C_{18}$

Related objects

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(27, 47);
 
Copy content sage:G = TransitiveGroup(27, 47)
 
Copy content oscar:G = transitive_group(27, 47)
 
Copy content gap:G := TransitiveGroup(27, 47);
 

Group invariants

Abstract group:  $C_3^2:C_{18}$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Copy content oscar:small_group_identification(G)
 
Copy content gap:IdGroup(G);
 
Order:  $162=2 \cdot 3^{4}$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Copy content gap:Order(G);
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content gap:IsCyclic(G);
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Copy content gap:IsAbelian(G);
 
Solvable:  yes
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Copy content gap:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 

Group action invariants

Degree $n$:  $27$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Copy content gap:NrMovedPoints(G);
 
Transitive number $t$:  $47$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
Copy content gap:TransitiveIdentification(G);
 
Parity:  $-1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Copy content gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
 
Transitivity:  1
Primitive:  no
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
Copy content gap:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $3$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(27).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(27), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(27), G));
 
Generators:  $(7,10,13)(8,11,14)(9,12,15)(16,23,21)(17,24,19)(18,22,20)$, $(1,8,22,2,9,23,3,7,24)(4,15,16,25,10,19,6,14,18,27,12,21,5,13,17,26,11,20)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$9$:  $C_9$
$18$:  $S_3\times C_3$, $C_{18}$
$54$:  $C_3^2 : C_6$, $C_9\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 9: $C_9$, $C_3^2 : S_3 $

Low degree siblings

18T82

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 $2^{9},1^{9}$ $9$ $2$ $9$ $( 4,27)( 5,25)( 6,26)(10,13)(11,14)(12,15)(16,21)(17,19)(18,20)$
3A1 $3^{9}$ $1$ $3$ $18$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)$
3A-1 $3^{9}$ $1$ $3$ $18$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)$
3B $3^{9}$ $2$ $3$ $18$ $( 1, 6,26)( 2, 4,27)( 3, 5,25)( 7,10,13)( 8,11,14)( 9,12,15)(16,21,23)(17,19,24)(18,20,22)$
3C1 $3^{9}$ $2$ $3$ $18$ $( 1, 5,27)( 2, 6,25)( 3, 4,26)( 7,12,14)( 8,10,15)( 9,11,13)(16,20,24)(17,21,22)(18,19,23)$
3C-1 $3^{9}$ $2$ $3$ $18$ $( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7,11,15)( 8,12,13)( 9,10,14)(16,19,22)(17,20,23)(18,21,24)$
3D $3^{6},1^{9}$ $6$ $3$ $12$ $( 1,26, 6)( 2,27, 4)( 3,25, 5)( 7,10,13)( 8,11,14)( 9,12,15)$
3E1 $3^{9}$ $6$ $3$ $18$ $( 1,25, 4)( 2,26, 5)( 3,27, 6)( 7,12,14)( 8,10,15)( 9,11,13)(16,18,17)(19,21,20)(22,24,23)$
3E-1 $3^{9}$ $6$ $3$ $18$ $( 1,27, 5)( 2,25, 6)( 3,26, 4)( 7,11,15)( 8,12,13)( 9,10,14)(16,17,18)(19,20,21)(22,23,24)$
6A1 $6^{3},3^{3}$ $9$ $6$ $21$ $( 1, 2, 3)( 4,25, 6,27, 5,26)( 7, 8, 9)(10,14,12,13,11,15)(16,19,18,21,17,20)(22,23,24)$
6A-1 $6^{3},3^{3}$ $9$ $6$ $21$ $( 1, 3, 2)( 4,26, 5,27, 6,25)( 7, 9, 8)(10,15,11,13,12,14)(16,20,17,21,18,19)(22,24,23)$
9A1 $9^{3}$ $3$ $9$ $24$ $( 1,22, 9, 3,24, 8, 2,23, 7)( 4,16,10, 6,18,12, 5,17,11)(13,26,20,15,25,19,14,27,21)$
9A-1 $9^{3}$ $3$ $9$ $24$ $( 1, 7,23, 2, 8,24, 3, 9,22)( 4,11,17, 5,12,18, 6,10,16)(13,21,27,14,19,25,15,20,26)$
9A2 $9^{3}$ $3$ $9$ $24$ $( 1, 9,24, 2, 7,22, 3, 8,23)( 4,10,18, 5,11,16, 6,12,17)(13,20,25,14,21,26,15,19,27)$
9A-2 $9^{3}$ $3$ $9$ $24$ $( 1,23, 8, 3,22, 7, 2,24, 9)( 4,17,12, 6,16,11, 5,18,10)(13,27,19,15,26,21,14,25,20)$
9A4 $9^{3}$ $3$ $9$ $24$ $( 1,24, 7, 3,23, 9, 2,22, 8)( 4,18,11, 6,17,10, 5,16,12)(13,25,21,15,27,20,14,26,19)$
9A-4 $9^{3}$ $3$ $9$ $24$ $( 1, 8,22, 2, 9,23, 3, 7,24)( 4,12,16, 5,10,17, 6,11,18)(13,19,26,14,20,27,15,21,25)$
9B1 $9^{3}$ $6$ $9$ $24$ $( 1,22,12, 3,24,11, 2,23,10)( 4,16,13, 6,18,15, 5,17,14)( 7,26,20, 9,25,19, 8,27,21)$
9B-1 $9^{3}$ $6$ $9$ $24$ $( 1,10,16, 2,11,17, 3,12,18)( 4,14,19, 5,15,20, 6,13,21)( 7,23,27, 8,24,25, 9,22,26)$
9B2 $9^{3}$ $6$ $9$ $24$ $( 1,12,17, 2,10,18, 3,11,16)( 4,13,20, 5,14,21, 6,15,19)( 7,22,25, 8,23,26, 9,24,27)$
9B-2 $9^{3}$ $6$ $9$ $24$ $( 1,23,11, 3,22,10, 2,24,12)( 4,17,15, 6,16,14, 5,18,13)( 7,27,19, 9,26,21, 8,25,20)$
9B4 $9^{3}$ $6$ $9$ $24$ $( 1,24,10, 3,23,12, 2,22,11)( 4,18,14, 6,17,13, 5,16,15)( 7,25,21, 9,27,20, 8,26,19)$
9B-4 $9^{3}$ $6$ $9$ $24$ $( 1,11,18, 2,12,16, 3,10,17)( 4,15,21, 5,13,19, 6,14,20)( 7,24,26, 8,22,27, 9,23,25)$
18A1 $18,9$ $9$ $18$ $25$ $( 1, 8,22, 2, 9,23, 3, 7,24)( 4,15,16,25,10,19, 6,14,18,27,12,21, 5,13,17,26,11,20)$
18A-1 $18,9$ $9$ $18$ $25$ $( 1,24, 7, 3,23, 9, 2,22, 8)( 4,20,11,26,17,13, 5,21,12,27,18,14, 6,19,10,25,16,15)$
18A5 $18,9$ $9$ $18$ $25$ $( 1,23, 8, 3,22, 7, 2,24, 9)( 4,19,12,26,16,14, 5,20,10,27,17,15, 6,21,11,25,18,13)$
18A-5 $18,9$ $9$ $18$ $25$ $( 1, 9,24, 2, 7,22, 3, 8,23)( 4,13,18,25,11,21, 6,15,17,27,10,20, 5,14,16,26,12,19)$
18A7 $18,9$ $9$ $18$ $25$ $( 1, 7,23, 2, 8,24, 3, 9,22)( 4,14,17,25,12,20, 6,13,16,27,11,19, 5,15,18,26,10,21)$
18A-7 $18,9$ $9$ $18$ $25$ $( 1,22, 9, 3,24, 8, 2,23, 7)( 4,21,10,26,18,15, 5,19,11,27,16,13, 6,20,12,25,17,14)$

Malle's constant $a(G)$:     $1/9$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 3B 3C1 3C-1 3D 3E1 3E-1 6A1 6A-1 9A1 9A-1 9A2 9A-2 9A4 9A-4 9B1 9B-1 9B2 9B-2 9B4 9B-4 18A1 18A-1 18A5 18A-5 18A7 18A-7
Size 1 9 1 1 2 2 2 6 6 6 9 9 3 3 3 3 3 3 6 6 6 6 6 6 9 9 9 9 9 9
2 P 1A 1A 3A-1 3A1 3B 3C-1 3C1 3D 3E-1 3E1 3A1 3A-1 9A2 9A-2 9A4 9A-4 9A-1 9A1 9B2 9B-2 9B4 9B-4 9B-1 9B1 9A1 9A-1 9A-4 9A4 9A-2 9A2
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 3A1 3A-1 3A-1 3A1 3A1 3A-1 3A1 3A-1 3A-1 3A1 3A1 3A-1 6A1 6A-1 6A-1 6A1 6A1 6A-1
Type
162.4.1a R 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 1
162.4.1b R 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 −1
162.4.1c1 C 1 1 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3 ζ3 ζ3−1 ζ3−1 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
162.4.1c2 C 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3−1 ζ3 ζ3 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
162.4.1d1 C 1 −1 1 1 1 1 1 1 1 1 −1 −1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3 ζ3 ζ3−1 ζ3−1 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1
162.4.1d2 C 1 −1 1 1 1 1 1 1 1 1 −1 −1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3−1 ζ3 ζ3 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3
162.4.1e1 C 1 1 ζ9−3 ζ93 ζ93 1 ζ9−3 ζ93 1 ζ9−3 ζ93 ζ9−3 ζ94 ζ9−4 ζ9−2 ζ9−1 ζ92 ζ9 ζ9−4 ζ92 ζ9−1 ζ9 ζ94 ζ9−2 ζ9−2 ζ9−4 ζ9−1 ζ9 ζ92 ζ94
162.4.1e2 C 1 1 ζ93 ζ9−3 ζ9−3 1 ζ93 ζ9−3 1 ζ93 ζ9−3 ζ93 ζ9−4 ζ94 ζ92 ζ9 ζ9−2 ζ9−1 ζ94 ζ9−2 ζ9 ζ9−1 ζ9−4 ζ92 ζ92 ζ94 ζ9 ζ9−1 ζ9−2 ζ9−4
162.4.1e3 C 1 1 ζ9−3 ζ93 ζ93 1 ζ9−3 ζ93 1 ζ9−3 ζ93 ζ9−3 ζ9−2 ζ92 ζ9 ζ9−4 ζ9−1 ζ94 ζ92 ζ9−1 ζ9−4 ζ94 ζ9−2 ζ9 ζ9 ζ92 ζ9−4 ζ94 ζ9−1 ζ9−2
162.4.1e4 C 1 1 ζ93 ζ9−3 ζ9−3 1 ζ93 ζ9−3 1 ζ93 ζ9−3 ζ93 ζ92 ζ9−2 ζ9−1 ζ94 ζ9 ζ9−4 ζ9−2 ζ9 ζ94 ζ9−4 ζ92 ζ9−1 ζ9−1 ζ9−2 ζ94 ζ9−4 ζ9 ζ92
162.4.1e5 C 1 1 ζ9−3 ζ93 ζ93 1 ζ9−3 ζ93 1 ζ9−3 ζ93 ζ9−3 ζ9 ζ9−1 ζ94 ζ92 ζ9−4 ζ9−2 ζ9−1 ζ9−4 ζ92 ζ9−2 ζ9 ζ94 ζ94 ζ9−1 ζ92 ζ9−2 ζ9−4 ζ9
162.4.1e6 C 1 1 ζ93 ζ9−3 ζ9−3 1 ζ93 ζ9−3 1 ζ93 ζ9−3 ζ93 ζ9−1 ζ9 ζ9−4 ζ9−2 ζ94 ζ92 ζ9 ζ94 ζ9−2 ζ92 ζ9−1 ζ9−4 ζ9−4 ζ9 ζ9−2 ζ92 ζ94 ζ9−1
162.4.1f1 C 1 −1 ζ9−3 ζ93 ζ93 1 ζ9−3 ζ93 1 ζ9−3 −ζ93 −ζ9−3 ζ94 ζ9−4 ζ9−2 ζ9−1 ζ92 ζ9 ζ9−4 ζ92 ζ9−1 ζ9 ζ94 ζ9−2 −ζ9−2 −ζ9−4 −ζ9−1 −ζ9 −ζ92 −ζ94
162.4.1f2 C 1 −1 ζ93 ζ9−3 ζ9−3 1 ζ93 ζ9−3 1 ζ93 −ζ9−3 −ζ93 ζ9−4 ζ94 ζ92 ζ9 ζ9−2 ζ9−1 ζ94 ζ9−2 ζ9 ζ9−1 ζ9−4 ζ92 −ζ92 −ζ94 −ζ9 −ζ9−1 −ζ9−2 −ζ9−4
162.4.1f3 C 1 −1 ζ9−3 ζ93 ζ93 1 ζ9−3 ζ93 1 ζ9−3 −ζ93 −ζ9−3 ζ9−2 ζ92 ζ9 ζ9−4 ζ9−1 ζ94 ζ92 ζ9−1 ζ9−4 ζ94 ζ9−2 ζ9 −ζ9 −ζ92 −ζ9−4 −ζ94 −ζ9−1 −ζ9−2
162.4.1f4 C 1 −1 ζ93 ζ9−3 ζ9−3 1 ζ93 ζ9−3 1 ζ93 −ζ9−3 −ζ93 ζ92 ζ9−2 ζ9−1 ζ94 ζ9 ζ9−4 ζ9−2 ζ9 ζ94 ζ9−4 ζ92 ζ9−1 −ζ9−1 −ζ9−2 −ζ94 −ζ9−4 −ζ9 −ζ92
162.4.1f5 C 1 −1 ζ9−3 ζ93 ζ93 1 ζ9−3 ζ93 1 ζ9−3 −ζ93 −ζ9−3 ζ9 ζ9−1 ζ94 ζ92 ζ9−4 ζ9−2 ζ9−1 ζ9−4 ζ92 ζ9−2 ζ9 ζ94 −ζ94 −ζ9−1 −ζ92 −ζ9−2 −ζ9−4 −ζ9
162.4.1f6 C 1 −1 ζ93 ζ9−3 ζ9−3 1 ζ93 ζ9−3 1 ζ93 −ζ9−3 −ζ93 ζ9−1 ζ9 ζ9−4 ζ9−2 ζ94 ζ92 ζ9 ζ94 ζ9−2 ζ92 ζ9−1 ζ9−4 −ζ9−4 −ζ9 −ζ9−2 −ζ92 −ζ94 −ζ9−1
162.4.2a R 2 0 2 2 2 2 2 −1 −1 −1 0 0 2 2 2 2 2 2 −1 −1 −1 −1 −1 −1 0 0 0 0 0 0
162.4.2b1 C 2 0 2 2 2 2 2 −1 −1 −1 0 0 2ζ3−1 2ζ3 2ζ3−1 2ζ3 2ζ3 2ζ3−1 −ζ3 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3−1 0 0 0 0 0 0
162.4.2b2 C 2 0 2 2 2 2 2 −1 −1 −1 0 0 2ζ3 2ζ3−1 2ζ3 2ζ3−1 2ζ3−1 2ζ3 −ζ3−1 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3 0 0 0 0 0 0
162.4.2c1 C 2 0 2ζ9−3 2ζ93 2ζ93 2 2ζ9−3 −ζ93 −1 −ζ9−3 0 0 2ζ94 2ζ9−4 2ζ9−2 2ζ9−1 2ζ92 2ζ9 −ζ9−4 −ζ92 −ζ9−1 −ζ9 −ζ94 −ζ9−2 0 0 0 0 0 0
162.4.2c2 C 2 0 2ζ93 2ζ9−3 2ζ9−3 2 2ζ93 −ζ9−3 −1 −ζ93 0 0 2ζ9−4 2ζ94 2ζ92 2ζ9 2ζ9−2 2ζ9−1 −ζ94 −ζ9−2 −ζ9 −ζ9−1 −ζ9−4 −ζ92 0 0 0 0 0 0
162.4.2c3 C 2 0 2ζ9−3 2ζ93 2ζ93 2 2ζ9−3 −ζ93 −1 −ζ9−3 0 0 2ζ9−2 2ζ92 2ζ9 2ζ9−4 2ζ9−1 2ζ94 −ζ92 −ζ9−1 −ζ9−4 −ζ94 −ζ9−2 −ζ9 0 0 0 0 0 0
162.4.2c4 C 2 0 2ζ93 2ζ9−3 2ζ9−3 2 2ζ93 −ζ9−3 −1 −ζ93 0 0 2ζ92 2ζ9−2 2ζ9−1 2ζ94 2ζ9 2ζ9−4 −ζ9−2 −ζ9 −ζ94 −ζ9−4 −ζ92 −ζ9−1 0 0 0 0 0 0
162.4.2c5 C 2 0 2ζ9−3 2ζ93 2ζ93 2 2ζ9−3 −ζ93 −1 −ζ9−3 0 0 2ζ9 2ζ9−1 2ζ94 2ζ92 2ζ9−4 2ζ9−2 −ζ9−1 −ζ9−4 −ζ92 −ζ9−2 −ζ9 −ζ94 0 0 0 0 0 0
162.4.2c6 C 2 0 2ζ93 2ζ9−3 2ζ9−3 2 2ζ93 −ζ9−3 −1 −ζ93 0 0 2ζ9−1 2ζ9 2ζ9−4 2ζ9−2 2ζ94 2ζ92 −ζ9 −ζ94 −ζ9−2 −ζ92 −ζ9−1 −ζ9−4 0 0 0 0 0 0
162.4.6a R 6 0 6 6 −3 −3 −3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
162.4.6b1 C 6 0 6ζ3−1 6ζ3 −3ζ3 −3 −3ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
162.4.6b2 C 6 0 6ζ3 6ζ3−1 −3ζ3−1 −3 −3ζ3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 
Copy content gap:CharacterTable(G);
 

Regular extensions

Data not computed