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Properties

Label 27T38
27T38 1 9 1->9 19 1->19 2 7 2->7 21 2->21 3 8 3->8 20 3->20 4 4->3 25 4->25 5 5->1 27 5->27 6 6->2 26 6->26 7->5 24 7->24 8->6 23 8->23 9->4 22 9->22 10 10->1 17 10->17 11 11->3 18 11->18 12 12->2 16 12->16 13 13->8 13->12 14 14->7 14->10 15 15->9 15->11 16->5 16->15 17->4 17->13 18->6 18->14 19->13 19->27 20->15 20->25 21->14 21->26 22->11 22->19 23->10 23->20 24->12 24->21 25->17 25->22 26->16 26->23 27->18 27->24
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, 38);
 
Copy content sage:G = TransitiveGroup(27, 38)
 
Copy content oscar:G = transitive_group(27, 38)
 
Copy content gap:G := TransitiveGroup(27, 38);
 

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 $t$:  $38$
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:  $(1,19,13,8,23,10)(2,21,14,7,24,12)(3,20,15,9,22,11)(4,25,17)(5,27,18,6,26,16)$, $(1,9,4,3,8,6,2,7,5)(10,17,13,12,16,15,11,18,14)(19,27,24,21,26,23,20,25,22)$
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

27T64

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 $2^{12},1^{3}$ $27$ $2$ $12$ $( 1, 4)( 2, 6)( 3, 5)( 7, 8)(10,11)(13,16)(14,18)(15,17)(19,20)(22,25)(23,27)(24,26)$
3A $3^{9}$ $2$ $3$ $18$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)$
3B $3^{6},1^{9}$ $6$ $3$ $12$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(19,21,20)(22,24,23)(25,27,26)$
3C1 $3^{9}$ $9$ $3$ $18$ $( 1,24,14)( 2,22,15)( 3,23,13)( 4,26,18)( 5,27,16)( 6,25,17)( 7,19,10)( 8,20,11)( 9,21,12)$
3C-1 $3^{9}$ $9$ $3$ $18$ $( 1,14,24)( 2,15,22)( 3,13,23)( 4,18,26)( 5,16,27)( 6,17,25)( 7,10,19)( 8,11,20)( 9,12,21)$
3D1 $3^{9}$ $18$ $3$ $18$ $( 1,20,17)( 2,21,18)( 3,19,16)( 4,22,10)( 5,23,11)( 6,24,12)( 7,26,13)( 8,27,14)( 9,25,15)$
3D-1 $3^{9}$ $18$ $3$ $18$ $( 1,17,20)( 2,18,21)( 3,16,19)( 4,10,22)( 5,11,23)( 6,12,24)( 7,13,26)( 8,14,27)( 9,15,25)$
6A1 $6^{4},3$ $27$ $6$ $22$ $( 1,18,24, 4,14,26)( 2,17,22, 6,15,25)( 3,16,23, 5,13,27)( 7,11,19, 8,10,20)( 9,12,21)$
6A-1 $6^{4},3$ $27$ $6$ $22$ $( 1,26,14, 4,24,18)( 2,25,15, 6,22,17)( 3,27,13, 5,23,16)( 7,20,10, 8,19,11)( 9,21,12)$
9A1 $9^{3}$ $6$ $9$ $24$ $( 1, 7, 6, 3, 9, 5, 2, 8, 4)(10,16,14,12,18,13,11,17,15)(19,27,24,21,26,23,20,25,22)$
9A2 $9^{3}$ $6$ $9$ $24$ $( 1, 6, 9, 2, 4, 7, 3, 5, 8)(10,14,18,11,15,16,12,13,17)(19,24,26,20,22,27,21,23,25)$
9A4 $9^{3}$ $6$ $9$ $24$ $( 1, 9, 4, 3, 8, 6, 2, 7, 5)(10,18,15,12,17,14,11,16,13)(19,26,22,21,25,24,20,27,23)$

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 3D1 3D-1 6A1 6A-1 9A1 9A2 9A4
Size 1 27 2 6 9 9 18 18 27 27 6 6 6
2 P 1A 1A 3A 3B 3C-1 3C1 3D-1 3D1 3C1 3C-1 9A2 9A4 9A1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 2A 2A 3A 3A 3A
Type
162.15.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
162.15.1b R 1 −1 1 1 1 1 1 1 −1 −1 1 1 1
162.15.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1
162.15.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1
162.15.1d1 C 1 −1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1
162.15.1d2 C 1 −1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1
162.15.2a R 2 0 2 2 2 2 −1 −1 0 0 −1 −1 −1
162.15.2b1 C 2 0 2 2 2ζ3−1 2ζ3 −ζ3−1 −ζ3 0 0 −1 −1 −1
162.15.2b2 C 2 0 2 2 2ζ3 2ζ3−1 −ζ3 −ζ3−1 0 0 −1 −1 −1
162.15.6a R 6 0 6 −3 0 0 0 0 0 0 0 0 0
162.15.6b1 R 6 0 −3 0 0 0 0 0 0 0 ζ9−4+ζ9−ζ92+2ζ94 −2ζ9−4+ζ9−ζ92−ζ94 ζ9−4−2ζ9+2ζ92−ζ94
162.15.6b2 R 6 0 −3 0 0 0 0 0 0 0 ζ9−4−2ζ9+2ζ92−ζ94 ζ9−4+ζ9−ζ92+2ζ94 −2ζ9−4+ζ9−ζ92−ζ94
162.15.6b3 R 6 0 −3 0 0 0 0 0 0 0 −2ζ9−4+ζ9−ζ92−ζ94 ζ9−4−2ζ9+2ζ92−ζ94 ζ9−4+ζ9−ζ92+2ζ94

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