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Properties

Label 27T29
27T29 1 9 1->9 26 1->26 2 8 2->8 25 2->25 3 7 3->7 27 3->27 4 4->1 15 4->15 5 5->3 14 5->14 6 6->2 13 6->13 16 7->16 19 7->19 18 8->18 21 8->21 17 9->17 20 9->20 10 22 10->22 24 10->24 11 23 11->23 11->24 12 12->22 12->23 13->17 13->19 14->16 14->21 15->18 15->20 16->6 16->9 17->5 17->8 18->4 18->7 19->3 19->11 20->2 20->10 21->1 21->12 22->13 22->27 23->15 23->26 24->14 24->25 25->6 25->12 26->5 26->11 27->4 27->10
Degree $27$
Order $108$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_3^2:D_6$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(27, 29);
 
Copy content sage:G = TransitiveGroup(27, 29)
 
Copy content oscar:G = transitive_group(27, 29)
 
Copy content gap:G := TransitiveGroup(27, 29);
 

Group invariants

Abstract group:  $C_3^2:D_6$
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 $t$:  $29$
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:  $(1,9,17,5,14,21)(2,8,18,4,15,20)(3,7,16,6,13,19)(10,22,27)(11,24,25,12,23,26)$, $(1,26,5,3,27,4)(2,25,6)(7,19,11,23,15,18)(8,21,12,22,13,17)(9,20,10,24,14,16)$
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 2
$12$:  $D_{6}$ x 2
$36$:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 2

Degree 9: $S_3^2$, $C_3^2 : D_{6} $ x 2

Low degree siblings

9T18 x 2, 18T51 x 2, 18T55 x 2, 18T56, 18T57 x 2, 36T87 x 2, 36T90

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

Malle's constant $a(G)$:     $1/12$

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 6A 6B 6C
Size 1 9 9 9 2 6 6 12 18 18 18
2 P 1A 1A 1A 1A 3A 3B 3C 3D 3B 3C 3A
3 P 1A 2A 2B 2C 1A 1A 1A 1A 2A 2B 2C
Type
108.17.1a R 1 1 1 1 1 1 1 1 1 1 1
108.17.1b R 1 −1 −1 1 1 1 1 1 1 −1 −1
108.17.1c R 1 −1 1 −1 1 1 1 1 −1 −1 1
108.17.1d R 1 1 −1 −1 1 1 1 1 −1 1 −1
108.17.2a R 2 0 2 0 2 2 −1 −1 0 0 −1
108.17.2b R 2 2 0 0 2 −1 2 −1 0 −1 0
108.17.2c R 2 −2 0 0 2 −1 2 −1 0 1 0
108.17.2d R 2 0 −2 0 2 2 −1 −1 0 0 1
108.17.4a R 4 0 0 0 4 −2 −2 1 0 0 0
108.17.6a R 6 0 0 2 −3 0 0 0 −1 0 0
108.17.6b R 6 0 0 −2 −3 0 0 0 1 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