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Properties

Label 27T34
27T34 1 2 1->2 19 1->19 26 1->26 3 2->3 20 2->20 25 2->25 3->1 21 3->21 27 3->27 4 4->1 5 4->5 17 4->17 5->3 6 5->6 18 5->18 6->2 6->4 16 6->16 7 7->6 8 7->8 24 7->24 8->4 9 8->9 23 8->23 9->5 9->7 22 9->22 10 10->3 11 10->11 10->18 11->1 12 11->12 11->17 12->2 12->10 12->16 13 14 13->14 13->21 13->27 15 14->15 14->20 14->25 15->13 15->19 15->26 16->13 16->14 16->17 17->15 17->15 17->18 18->13 18->14 18->16 19->8 19->11 19->20 20->7 20->12 20->21 21->9 21->10 21->19 22->7 22->12 22->23 23->8 23->11 23->24 24->9 24->10 24->22 25->5 25->22 25->26 26->4 26->23 26->27 27->6 27->24 27->25
Degree $27$
Order $108$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_3:S_3^2$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(27, 34);
 
Copy content sage:G = TransitiveGroup(27, 34)
 
Copy content oscar:G = transitive_group(27, 34)
 
Copy content gap:G := TransitiveGroup(27, 34);
 

Group invariants

Abstract group:  $C_3:S_3^2$
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:  $108=2^{2} \cdot 3^{3}$
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 $t$:  $34$
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);
 
$\card{\Aut(F/K)}$:  $1$
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:  $(1,19,11)(2,20,12)(3,21,10)(4,17,15,26,23,8)(5,18,13,27,24,9)(6,16,14,25,22,7)$, $(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)$, $(1,26,4)(2,25,5,3,27,6)(7,24,10,18,14,20)(8,23,11,17,15,19)(9,22,12,16,13,21)$
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)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 5
$12$:  $D_{6}$ x 5
$18$:  $C_3^2:C_2$
$36$:  $S_3^2$ x 4, 18T12

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 5

Degree 9: $C_3^2:C_2$, $S_3^2$ x 4

Low degree siblings

18T58 x 4, 36T91 x 4

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

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 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 3I 6A 6B 6C 6D 6E
Size 1 3 9 27 2 2 2 2 2 4 4 4 4 6 6 6 6 18
2 P 1A 1A 1A 1A 3A 3B 3C 3D 3E 3F 3G 3H 3I 3A 3B 3C 3D 3E
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2B
Type
108.39.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
108.39.1b R 1 −1 −1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1
108.39.1c R 1 −1 1 −1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1
108.39.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1
108.39.2a R 2 0 2 0 2 2 2 2 −1 −1 −1 −1 −1 0 0 0 0 −1
108.39.2b R 2 2 0 0 −1 −1 −1 2 2 −1 −1 −1 2 −1 −1 −1 2 0
108.39.2c R 2 2 0 0 −1 −1 2 −1 2 −1 −1 2 −1 −1 −1 2 −1 0
108.39.2d R 2 2 0 0 −1 2 −1 −1 2 −1 2 −1 −1 −1 2 −1 −1 0
108.39.2e R 2 2 0 0 2 −1 −1 −1 2 2 −1 −1 −1 2 −1 −1 −1 0
108.39.2f R 2 −2 0 0 −1 −1 −1 2 2 −1 −1 −1 2 1 1 1 −2 0
108.39.2g R 2 −2 0 0 −1 −1 2 −1 2 −1 −1 2 −1 1 1 −2 1 0
108.39.2h R 2 −2 0 0 −1 2 −1 −1 2 −1 2 −1 −1 1 −2 1 1 0
108.39.2i R 2 −2 0 0 2 −1 −1 −1 2 2 −1 −1 −1 −2 1 1 1 0
108.39.2j R 2 0 −2 0 2 2 2 2 −1 −1 −1 −1 −1 0 0 0 0 1
108.39.4a R 4 0 0 0 −2 −2 −2 4 −2 1 1 1 −2 0 0 0 0 0
108.39.4b R 4 0 0 0 −2 −2 4 −2 −2 1 1 −2 1 0 0 0 0 0
108.39.4c R 4 0 0 0 −2 4 −2 −2 −2 1 −2 1 1 0 0 0 0 0
108.39.4d R 4 0 0 0 4 −2 −2 −2 −2 −2 1 1 1 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