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Properties

Label 27T141
27T141 1 4 1->4 23 1->23 2 6 2->6 22 2->22 3 5 3->5 24 3->24 7 4->7 9 4->9 8 5->8 5->9 6->7 6->8 12 7->12 26 7->26 11 8->11 25 8->25 10 9->10 27 9->27 20 10->20 10->24 19 11->19 11->23 21 12->21 12->22 13 15 13->15 13->25 14 14->27 15->26 16 16->3 16->4 17 17->2 17->6 18 18->1 18->5 19->14 19->18 20->13 20->17 21->15 21->16 22->10 22->17 23->12 23->16 24->11 24->18 25->2 25->20 26->1 26->19 27->3 27->21
Degree $27$
Order $432$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $\He_3:\SD_{16}$

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, 141);
 
Copy content sage:G = TransitiveGroup(27, 141)
 
Copy content oscar:G = transitive_group(27, 141)
 
Copy content gap:G := TransitiveGroup(27, 141);
 

Group invariants

Abstract group:  $\He_3:\SD_{16}$
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:  $432=2^{4} \cdot 3^{3}$
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 $t$:  $141$
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:  $(1,23,12,21,16,4,7,26)(2,22,10,20,17,6,8,25)(3,24,11,19,18,5,9,27)(13,15)$, $(1,4,9,10,24,18)(2,6,7,12,22,17)(3,5,8,11,23,16)(13,25,20)(14,27,21,15,26,19)$
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
$4$:  $C_2^2$
$8$:  $D_{4}$
$16$:  $QD_{16}$
$144$:  $(C_3^2:C_8):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 9: $(C_3^2:C_8):C_2$

Low degree siblings

36T704

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 $2^{12},1^{3}$ $9$ $2$ $12$ $( 1, 8)( 2, 9)( 3, 7)( 4,14)( 5,15)( 6,13)(10,25)(11,26)(12,27)(16,20)(17,21)(18,19)$
2B $2^{12},1^{3}$ $36$ $2$ $12$ $( 1, 3)( 4, 5)( 7,19)( 8,21)( 9,20)(10,22)(11,24)(12,23)(13,16)(14,18)(15,17)(25,27)$
3A $3^{9}$ $2$ $3$ $18$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)$
3B $3^{9}$ $24$ $3$ $18$ $( 1,25, 5)( 2,26, 6)( 3,27, 4)( 7,15,11)( 8,13,12)( 9,14,10)(16,23,21)(17,24,19)(18,22,20)$
4A $4^{6},1^{3}$ $18$ $4$ $18$ $( 1, 4,19,17)( 2, 5,20,18)( 3, 6,21,16)( 7,22,14,27)( 8,23,15,25)( 9,24,13,26)$
4B $4^{6},1^{3}$ $36$ $4$ $18$ $( 1,27, 8,12)( 2,25, 9,10)( 3,26, 7,11)( 4,18,14,19)( 5,16,15,20)( 6,17,13,21)$
6A $6^{4},3$ $18$ $6$ $22$ $( 1, 7, 2, 8, 3, 9)( 4,13, 5,14, 6,15)(10,27,11,25,12,26)(16,19,17,20,18,21)(22,24,23)$
6B $6^{4},3$ $72$ $6$ $22$ $( 1, 4,25, 3, 5,27)( 2, 6,26)( 7,24,15,19,11,17)( 8,23,13,21,12,16)( 9,22,14,20,10,18)$
8A1 $8^{3},2,1$ $54$ $8$ $22$ $( 1,13, 4,26,19, 9,17,24)( 2,15, 5,25,20, 8,18,23)( 3,14, 6,27,21, 7,16,22)(10,12)$
8A-1 $8^{3},2,1$ $54$ $8$ $22$ $( 1,24,17, 9,19,26, 4,13)( 2,23,18, 8,20,25, 5,15)( 3,22,16, 7,21,27, 6,14)(10,12)$
12A $12^{2},3$ $36$ $12$ $24$ $( 1,14,12,27, 3,13,11,26, 2,15,10,25)( 4,18, 9,24, 6,17, 8,23, 5,16, 7,22)(19,21,20)$
12B1 $12^{2},3$ $36$ $12$ $24$ $( 1,10, 7,27, 2,11, 8,25, 3,12, 9,26)( 4,20,13,18, 5,21,14,16, 6,19,15,17)(22,23,24)$
12B-1 $12^{2},3$ $36$ $12$ $24$ $( 1,26, 9,12, 3,25, 8,11, 2,27, 7,10)( 4,17,15,19, 6,16,14,21, 5,18,13,20)(22,24,23)$

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 3A 3B 4A 4B 6A 6B 8A1 8A-1 12A 12B1 12B-1
Size 1 9 36 2 24 18 36 18 72 54 54 36 36 36
2 P 1A 1A 1A 3A 3B 2A 2A 3A 3B 4A 4A 6A 6A 6A
3 P 1A 2A 2B 1A 1A 4A 4B 2A 2B 8A1 8A-1 4A 4B 4B
Type
432.520.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
432.520.1b R 1 1 −1 1 1 1 −1 1 −1 1 1 −1 −1 1
432.520.1c R 1 1 −1 1 1 1 1 1 −1 −1 −1 1 1 1
432.520.1d R 1 1 1 1 1 1 −1 1 1 −1 −1 −1 −1 1
432.520.2a R 2 2 0 2 2 −2 0 2 0 0 0 0 0 −2
432.520.2b1 C 2 −2 0 2 2 0 0 −2 0 −ζ8−ζ83 ζ8+ζ83 0 0 0
432.520.2b2 C 2 −2 0 2 2 0 0 −2 0 ζ8+ζ83 −ζ8−ζ83 0 0 0
432.520.6a R 6 −2 0 −3 0 2 2 1 0 0 0 −1 −1 −1
432.520.6b R 6 −2 0 −3 0 2 −2 1 0 0 0 1 1 −1
432.520.6c1 C 6 −2 0 −3 0 −2 0 1 0 0 0 −1−2ζ3 1+2ζ3 1
432.520.6c2 C 6 −2 0 −3 0 −2 0 1 0 0 0 1+2ζ3 −1−2ζ3 1
432.520.8a R 8 0 2 8 −1 0 0 0 −1 0 0 0 0 0
432.520.8b R 8 0 −2 8 −1 0 0 0 1 0 0 0 0 0
432.520.12a R 12 4 0 −6 0 0 0 −2 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