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Properties

Label 27T28
27T28 1 4 1->4 16 1->16 2 5 2->5 17 2->17 3 6 3->6 18 3->18 7 4->7 10 4->10 8 5->8 11 5->11 9 6->9 12 6->12 7->2 14 7->14 8->3 15 8->15 9->1 13 9->13 10->11 10->13 23 10->23 11->12 11->14 24 11->24 12->10 12->15 22 12->22 13->14 13->18 27 13->27 14->15 14->16 25 14->25 15->13 15->17 26 15->26 16->12 16->17 19 16->19 17->10 17->18 20 17->20 18->11 18->16 21 18->21 19->1 19->21 19->24 20->2 20->19 20->22 21->3 21->20 21->23 22->6 22->24 22->25 23->4 23->22 23->26 24->5 24->23 24->27 25->7 25->21 25->27 26->8 26->19 26->25 27->9 27->20 27->26
Degree $27$
Order $81$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_9.C_3^2$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(27, 28);
 
Copy content sage:G = TransitiveGroup(27, 28)
 
Copy content oscar:G = transitive_group(27, 28)
 
Copy content gap:G := TransitiveGroup(27, 28);
 

Group invariants

Abstract group:  $C_9.C_3^2$
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:  $81=3^{4}$
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:  $2$
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 $t$:  $28$
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);
 
$\card{\Aut(F/K)}$:  $9$
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:  $(1,16,19)(2,17,20)(3,18,21)(4,10,23)(5,11,24)(6,12,22)(7,14,25)(8,15,26)(9,13,27)$, $(1,4,7,2,5,8,3,6,9)(10,13,18,11,14,16,12,15,17)(19,24,27,20,22,25,21,23,26)$, $(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)$
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)
$3$:  $C_3$ x 13
$9$:  $C_3^2$ x 13
$27$:  27T4

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$ x 4

Degree 9: $C_3^2$

Low degree siblings

27T28 x 3

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{27}$ $1$ $1$ $0$ $()$
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 $3^{9}$ $1$ $3$ $18$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)$
3B1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)$
3B-1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,14,15)(16,17,18)$
3C1 $3^{9}$ $3$ $3$ $18$ $( 1,19,18)( 2,20,16)( 3,21,17)( 4,23,12)( 5,24,10)( 6,22,11)( 7,25,13)( 8,26,14)( 9,27,15)$
3C-1 $3^{9}$ $3$ $3$ $18$ $( 1,18,19)( 2,16,20)( 3,17,21)( 4,12,23)( 5,10,24)( 6,11,22)( 7,13,25)( 8,14,26)( 9,15,27)$
3D1 $3^{9}$ $3$ $3$ $18$ $( 1,19,17)( 2,20,18)( 3,21,16)( 4,23,11)( 5,24,12)( 6,22,10)( 7,25,15)( 8,26,13)( 9,27,14)$
3D-1 $3^{9}$ $3$ $3$ $18$ $( 1,16,20)( 2,17,21)( 3,18,19)( 4,10,24)( 5,11,22)( 6,12,23)( 7,14,26)( 8,15,27)( 9,13,25)$
3E1 $3^{9}$ $3$ $3$ $18$ $( 1,19,16)( 2,20,17)( 3,21,18)( 4,23,10)( 5,24,11)( 6,22,12)( 7,25,14)( 8,26,15)( 9,27,13)$
3E-1 $3^{9}$ $3$ $3$ $18$ $( 1,17,21)( 2,18,19)( 3,16,20)( 4,11,22)( 5,12,23)( 6,10,24)( 7,15,27)( 8,13,25)( 9,14,26)$
9A1 $9^{3}$ $1$ $9$ $24$ $( 1, 4, 7, 2, 5, 8, 3, 6, 9)(10,14,17,11,15,18,12,13,16)(19,23,25,20,24,26,21,22,27)$
9A-1 $9^{3}$ $1$ $9$ $24$ $( 1, 9, 6, 3, 8, 5, 2, 7, 4)(10,16,13,12,18,15,11,17,14)(19,27,22,21,26,24,20,25,23)$
9A2 $9^{3}$ $1$ $9$ $24$ $( 1, 7, 5, 3, 9, 4, 2, 8, 6)(10,17,15,12,16,14,11,18,13)(19,25,24,21,27,23,20,26,22)$
9A-2 $9^{3}$ $1$ $9$ $24$ $( 1, 6, 8, 2, 4, 9, 3, 5, 7)(10,13,18,11,14,16,12,15,17)(19,22,26,20,23,27,21,24,25)$
9A4 $9^{3}$ $1$ $9$ $24$ $( 1, 5, 9, 2, 6, 7, 3, 4, 8)(10,15,16,11,13,17,12,14,18)(19,24,27,20,22,25,21,23,26)$
9A-4 $9^{3}$ $1$ $9$ $24$ $( 1, 8, 4, 3, 7, 6, 2, 9, 5)(10,18,14,12,17,13,11,16,15)(19,26,23,21,25,22,20,27,24)$
9B1 $9^{3}$ $3$ $9$ $24$ $( 1, 5, 9, 2, 6, 7, 3, 4, 8)(10,13,18,11,14,16,12,15,17)(19,23,25,20,24,26,21,22,27)$
9B-1 $9^{3}$ $3$ $9$ $24$ $( 1, 9, 6, 3, 8, 5, 2, 7, 4)(10,18,14,12,17,13,11,16,15)(19,25,24,21,27,23,20,26,22)$
9C1 $9^{3}$ $3$ $9$ $24$ $( 1,23,13, 2,24,14, 3,22,15)( 4,25,16, 5,26,17, 6,27,18)( 7,20,10, 8,21,11, 9,19,12)$
9C-1 $9^{3}$ $3$ $9$ $24$ $( 1,13,24, 3,15,23, 2,14,22)( 4,16,26, 6,18,25, 5,17,27)( 7,10,21, 9,12,20, 8,11,19)$
9D1 $9^{3}$ $3$ $9$ $24$ $( 1, 8, 4, 3, 7, 6, 2, 9, 5)(10,16,13,12,18,15,11,17,14)(19,25,24,21,27,23,20,26,22)$
9D-1 $9^{3}$ $3$ $9$ $24$ $( 1, 6, 8, 2, 4, 9, 3, 5, 7)(10,15,16,11,13,17,12,14,18)(19,23,25,20,24,26,21,22,27)$
9E1 $9^{3}$ $3$ $9$ $24$ $( 1,25,10, 3,27,12, 2,26,11)( 4,20,14, 6,19,13, 5,21,15)( 7,24,17, 9,23,16, 8,22,18)$
9E-1 $9^{3}$ $3$ $9$ $24$ $( 1,12,25, 2,10,26, 3,11,27)( 4,13,20, 5,14,21, 6,15,19)( 7,16,24, 8,17,22, 9,18,23)$
9F1 $9^{3}$ $3$ $9$ $24$ $( 1,23,15, 2,24,13, 3,22,14)( 4,25,18, 5,26,16, 6,27,17)( 7,20,12, 8,21,10, 9,19,11)$
9F-1 $9^{3}$ $3$ $9$ $24$ $( 1,14,22, 3,13,24, 2,15,23)( 4,17,27, 6,16,26, 5,18,25)( 7,11,19, 9,10,21, 8,12,20)$
9G1 $9^{3}$ $3$ $9$ $24$ $( 1,25,12, 3,27,11, 2,26,10)( 4,20,13, 6,19,15, 5,21,14)( 7,24,16, 9,23,18, 8,22,17)$
9G-1 $9^{3}$ $3$ $9$ $24$ $( 1,10,26, 2,11,27, 3,12,25)( 4,14,21, 5,15,19, 6,13,20)( 7,17,22, 8,18,23, 9,16,24)$
9H1 $9^{3}$ $3$ $9$ $24$ $( 1,23,14, 2,24,15, 3,22,13)( 4,25,17, 5,26,18, 6,27,16)( 7,20,11, 8,21,12, 9,19,10)$
9H-1 $9^{3}$ $3$ $9$ $24$ $( 1,15,23, 3,14,22, 2,13,24)( 4,18,25, 6,17,27, 5,16,26)( 7,12,20, 9,11,19, 8,10,21)$
9I1 $9^{3}$ $3$ $9$ $24$ $( 1,11,27, 2,12,25, 3,10,26)( 4,15,19, 5,13,20, 6,14,21)( 7,18,23, 8,16,24, 9,17,22)$
9I-1 $9^{3}$ $3$ $9$ $24$ $( 1,25,11, 3,27,10, 2,26,12)( 4,20,15, 6,19,14, 5,21,13)( 7,24,18, 9,23,17, 8,22,16)$

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

33 x 33 character table

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

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