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Properties

Label 27T41
27T41 1 10 1->10 20 1->20 2 11 2->11 19 2->19 3 12 3->12 21 3->21 4 15 4->15 26 4->26 5 13 5->13 25 5->25 6 14 6->14 27 6->27 7 16 7->16 22 7->22 8 17 8->17 24 8->24 9 18 9->18 23 9->23 10->1 10->25 11->3 11->26 12->2 12->27 13->9 13->20 14->8 14->21 15->7 15->19 16->4 16->23 17->6 17->24 18->5 18->22 19->5 19->15 20->6 20->14 21->4 21->13 22->7 22->12 23->8 23->11 24->9 24->10 25->2 25->17 26->3 26->16 27->1 27->18
Degree $27$
Order $162$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $\He_3.S_3$

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, 41);
 
Copy content sage:G = TransitiveGroup(27, 41)
 
Copy content oscar:G = transitive_group(27, 41)
 
Copy content gap:G := TransitiveGroup(27, 41);
 

Group invariants

Abstract group:  $\He_3.S_3$
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 $t$:  $41$
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:  $(1,10,25,2,11,26,3,12,27)(4,15,19,5,13,20,6,14,21)(7,16,23,8,17,24,9,18,22)$, $(1,20,14,8,24,10)(2,19,15,7,22,12)(3,21,13,9,23,11)(4,26,16)(5,25,17,6,27,18)$
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)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$
$54$:  $C_3^2 : C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$, $S_3$

Degree 9: $S_3\times C_3$

Low degree siblings

27T72

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

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 3A 3B 3C1 3C-1 6A1 6A-1 9A1 9A2 9A4 9B1 9B-1
Size 1 27 2 6 9 9 27 27 6 6 6 18 18
2 P 1A 1A 3A 3B 3C-1 3C1 3C1 3C-1 9A2 9A4 9A1 9B-1 9B1
3 P 1A 2A 1A 1A 1A 1A 2A 2A 3A 3A 3A 3A 3A
Type
162.13.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
162.13.1b R 1 −1 1 1 1 1 −1 −1 1 1 1 1 1
162.13.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3
162.13.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1
162.13.1d1 C 1 −1 1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 ζ3−1 ζ3
162.13.1d2 C 1 −1 1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 ζ3 ζ3−1
162.13.2a R 2 0 2 2 2 2 0 0 −1 −1 −1 −1 −1
162.13.2b1 C 2 0 2 2 2ζ3−1 2ζ3 0 0 −1 −1 −1 −ζ3−1 −ζ3
162.13.2b2 C 2 0 2 2 2ζ3 2ζ3−1 0 0 −1 −1 −1 −ζ3 −ζ3−1
162.13.6a R 6 0 6 −3 0 0 0 0 0 0 0 0 0
162.13.6b1 R 6 0 −3 0 0 0 0 0 2ζ9−4−ζ9+ζ92+ζ94 −ζ9−4+2ζ9−2ζ92+ζ94 −ζ9−4−ζ9+ζ92−2ζ94 0 0
162.13.6b2 R 6 0 −3 0 0 0 0 0 −ζ9−4+2ζ9−2ζ92+ζ94 −ζ9−4−ζ9+ζ92−2ζ94 2ζ9−4−ζ9+ζ92+ζ94 0 0
162.13.6b3 R 6 0 −3 0 0 0 0 0 −ζ9−4−ζ9+ζ92−2ζ94 2ζ9−4−ζ9+ζ92+ζ94 −ζ9−4+2ζ9−2ζ92+ζ94 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