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Properties

Label 27T40
27T40 1 23 1->23 25 1->25 2 24 2->24 26 2->26 3 22 3->22 27 3->27 4 21 4->21 4->26 5 19 5->19 5->27 6 20 6->20 6->25 7 7->21 7->22 8 8->19 8->23 9 9->20 9->24 10 10->8 11 11->9 12 12->7 13 13->2 14 14->3 15 15->1 16 16->6 17 17->4 18 18->5 19->10 20->11 21->12 22->14 23->15 24->13 25->16 26->17 27->18
Degree $27$
Order $162$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $\He_3.C_6$

Related objects

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(27, 40);
 
Copy content sage:G = TransitiveGroup(27, 40)
 
Copy content oscar:G = transitive_group(27, 40)
 
Copy content gap:G := TransitiveGroup(27, 40);
 

Group invariants

Abstract group:  $\He_3.C_6$
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:  $162=2 \cdot 3^{4}$
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 $t$:  $40$
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:  $(1,23,15)(2,24,13)(3,22,14)(4,26,17)(5,27,18)(6,25,16)(7,21,12)(8,19,10)(9,20,11)$, $(1,25)(2,26)(3,27)(4,21)(5,19)(6,20)(7,22)(8,23)(9,24)$
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

27T49

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 $2^{9},1^{9}$ $9$ $2$ $9$ $(10,27)(11,25)(12,26)(13,20)(14,21)(15,19)(16,23)(17,24)(18,22)$
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 $3^{9}$ $1$ $3$ $18$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)$
3B $3^{6},1^{9}$ $6$ $3$ $12$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(19,20,21)(22,23,24)(25,26,27)$
3C $3^{9}$ $18$ $3$ $18$ $( 1,11,26)( 2,12,27)( 3,10,25)( 4,14,19)( 5,15,20)( 6,13,21)( 7,18,23)( 8,16,24)( 9,17,22)$
3D1 $3^{9}$ $18$ $3$ $18$ $( 1,13,22)( 2,14,23)( 3,15,24)( 4,18,25)( 5,16,26)( 6,17,27)( 7,10,20)( 8,11,21)( 9,12,19)$
3D-1 $3^{9}$ $18$ $3$ $18$ $( 1,22,13)( 2,23,14)( 3,24,15)( 4,25,18)( 5,26,16)( 6,27,17)( 7,20,10)( 8,21,11)( 9,19,12)$
6A1 $6^{3},3^{3}$ $9$ $6$ $21$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,26,11,27,12,25)(13,19,14,20,15,21)(16,22,17,23,18,24)$
6A-1 $6^{3},3^{3}$ $9$ $6$ $21$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,25,12,27,11,26)(13,21,15,20,14,19)(16,24,18,23,17,22)$
9A1 $9^{3}$ $3$ $9$ $24$ $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,14,17,11,15,18,12,13,16)(19,22,26,20,23,27,21,24,25)$
9A-1 $9^{3}$ $3$ $9$ $24$ $( 1, 7, 6, 3, 9, 5, 2, 8, 4)(10,16,13,12,18,15,11,17,14)(19,25,24,21,27,23,20,26,22)$
9A2 $9^{3}$ $3$ $9$ $24$ $( 1, 8, 5, 3, 7, 4, 2, 9, 6)(10,17,15,12,16,14,11,18,13)(19,26,23,21,25,22,20,27,24)$
9A-2 $9^{3}$ $3$ $9$ $24$ $( 1, 6, 9, 2, 4, 7, 3, 5, 8)(10,13,18,11,14,16,12,15,17)(19,24,27,20,22,25,21,23,26)$
9A4 $9^{3}$ $3$ $9$ $24$ $( 1, 5, 7, 2, 6, 8, 3, 4, 9)(10,15,16,11,13,17,12,14,18)(19,23,25,20,24,26,21,22,27)$
9A-4 $9^{3}$ $3$ $9$ $24$ $( 1, 9, 4, 3, 8, 6, 2, 7, 5)(10,18,14,12,17,13,11,16,15)(19,27,22,21,26,24,20,25,23)$
18A1 $18,9$ $9$ $18$ $25$ $( 1, 9, 4, 3, 8, 6, 2, 7, 5)(10,22,14,26,17,20,11,23,15,27,18,21,12,24,13,25,16,19)$
18A-1 $18,9$ $9$ $18$ $25$ $( 1,13, 9,12, 4,18, 3,15, 8,11, 6,17, 2,14, 7,10, 5,16)(19,22,26,20,23,27,21,24,25)$
18A5 $18,9$ $9$ $18$ $25$ $( 1, 6, 9, 2, 4, 7, 3, 5, 8)(10,20,18,25,14,23,12,19,17,27,13,22,11,21,16,26,15,24)$
18A-5 $18,9$ $9$ $18$ $25$ $( 1, 8, 5, 3, 7, 4, 2, 9, 6)(10,24,15,26,16,21,11,22,13,27,17,19,12,23,14,25,18,20)$
18A7 $18,9$ $9$ $18$ $25$ $( 1, 7, 6, 3, 9, 5, 2, 8, 4)(10,23,13,26,18,19,11,24,14,27,16,20,12,22,15,25,17,21)$
18A-7 $18,9$ $9$ $18$ $25$ $( 1, 4, 8, 2, 5, 9, 3, 6, 7)(10,21,17,25,15,22,12,20,16,27,14,24,11,19,18,26,13,23)$

Malle's constant $a(G)$:     $1/9$

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 3A1 3A-1 3B 3C 3D1 3D-1 6A1 6A-1 9A1 9A-1 9A2 9A-2 9A4 9A-4 18A1 18A-1 18A5 18A-5 18A7 18A-7
Size 1 9 1 1 6 18 18 18 9 9 3 3 3 3 3 3 9 9 9 9 9 9
2 P 1A 1A 3A-1 3A1 3B 3C 3D-1 3D1 3A1 3A-1 9A2 9A-2 9A4 9A-4 9A-1 9A1 9A1 9A-1 9A-4 9A4 9A-2 9A2
3 P 1A 2A 1A 1A 1A 1A 1A 1A 2A 2A 3A1 3A-1 3A-1 3A1 3A1 3A-1 6A1 6A-1 6A-1 6A1 6A1 6A-1
Type
162.14.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
162.14.1b R 1 −1 1 1 1 1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
162.14.1c1 C 1 1 1 1 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3−1 ζ3 ζ3
162.14.1c2 C 1 1 1 1 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3 ζ3−1 ζ3−1
162.14.1d1 C 1 −1 1 1 1 1 ζ3−1 ζ3 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3−1 −ζ3 −ζ3
162.14.1d2 C 1 −1 1 1 1 1 ζ3 ζ3−1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3 −ζ3−1 −ζ3−1
162.14.2a R 2 0 2 2 2 −1 −1 −1 0 0 2 2 2 2 2 2 0 0 0 0 0 0
162.14.2b1 C 2 0 2 2 2 −1 −ζ3 −ζ3−1 0 0 2ζ3 2ζ3−1 2ζ3−1 2ζ3 2ζ3 2ζ3−1 0 0 0 0 0 0
162.14.2b2 C 2 0 2 2 2 −1 −ζ3−1 −ζ3 0 0 2ζ3−1 2ζ3 2ζ3 2ζ3−1 2ζ3−1 2ζ3 0 0 0 0 0 0
162.14.3a1 C 3 1 3ζ9−3 3ζ93 0 0 0 0 ζ93 ζ9−3 ζ9−4−ζ92 ζ9+2ζ94 ζ9−ζ94 −2ζ9−4−ζ92 ζ9−4+2ζ92 −2ζ9−ζ94 ζ94 ζ9−1 ζ92 ζ9−4 ζ9 ζ9−2
162.14.3a2 C 3 1 3ζ93 3ζ9−3 0 0 0 0 ζ9−3 ζ93 ζ9+2ζ94 ζ9−4−ζ92 −2ζ9−4−ζ92 ζ9−ζ94 −2ζ9−ζ94 ζ9−4+2ζ92 ζ9−4 ζ9 ζ9−2 ζ94 ζ9−1 ζ92
162.14.3a3 C 3 1 3ζ9−3 3ζ93 0 0 0 0 ζ93 ζ9−3 ζ9−4+2ζ92 −2ζ9−ζ94 ζ9+2ζ94 ζ9−4−ζ92 −2ζ9−4−ζ92 ζ9−ζ94 ζ9−2 ζ9−4 ζ9−1 ζ92 ζ94 ζ9
162.14.3a4 C 3 1 3ζ93 3ζ9−3 0 0 0 0 ζ9−3 ζ93 −2ζ9−ζ94 ζ9−4+2ζ92 ζ9−4−ζ92 ζ9+2ζ94 ζ9−ζ94 −2ζ9−4−ζ92 ζ92 ζ94 ζ9 ζ9−2 ζ9−4 ζ9−1
162.14.3a5 C 3 1 3ζ9−3 3ζ93 0 0 0 0 ζ93 ζ9−3 −2ζ9−4−ζ92 ζ9−ζ94 −2ζ9−ζ94 ζ9−4+2ζ92 ζ9−4−ζ92 ζ9+2ζ94 ζ9 ζ92 ζ9−4 ζ9−1 ζ9−2 ζ94
162.14.3a6 C 3 1 3ζ93 3ζ9−3 0 0 0 0 ζ9−3 ζ93 ζ9−ζ94 −2ζ9−4−ζ92 ζ9−4+2ζ92 −2ζ9−ζ94 ζ9+2ζ94 ζ9−4−ζ92 ζ9−1 ζ9−2 ζ94 ζ9 ζ92 ζ9−4
162.14.3b1 C 3 −1 3ζ9−3 3ζ93 0 0 0 0 −ζ93 −ζ9−3 ζ9−4−ζ92 ζ9+2ζ94 ζ9−ζ94 −2ζ9−4−ζ92 ζ9−4+2ζ92 −2ζ9−ζ94 −ζ94 −ζ9−1 −ζ92 −ζ9−4 −ζ9 −ζ9−2
162.14.3b2 C 3 −1 3ζ93 3ζ9−3 0 0 0 0 −ζ9−3 −ζ93 ζ9+2ζ94 ζ9−4−ζ92 −2ζ9−4−ζ92 ζ9−ζ94 −2ζ9−ζ94 ζ9−4+2ζ92 −ζ9−4 −ζ9 −ζ9−2 −ζ94 −ζ9−1 −ζ92
162.14.3b3 C 3 −1 3ζ9−3 3ζ93 0 0 0 0 −ζ93 −ζ9−3 ζ9−4+2ζ92 −2ζ9−ζ94 ζ9+2ζ94 ζ9−4−ζ92 −2ζ9−4−ζ92 ζ9−ζ94 −ζ9−2 −ζ9−4 −ζ9−1 −ζ92 −ζ94 −ζ9
162.14.3b4 C 3 −1 3ζ93 3ζ9−3 0 0 0 0 −ζ9−3 −ζ93 −2ζ9−ζ94 ζ9−4+2ζ92 ζ9−4−ζ92 ζ9+2ζ94 ζ9−ζ94 −2ζ9−4−ζ92 −ζ92 −ζ94 −ζ9 −ζ9−2 −ζ9−4 −ζ9−1
162.14.3b5 C 3 −1 3ζ9−3 3ζ93 0 0 0 0 −ζ93 −ζ9−3 −2ζ9−4−ζ92 ζ9−ζ94 −2ζ9−ζ94 ζ9−4+2ζ92 ζ9−4−ζ92 ζ9+2ζ94 −ζ9 −ζ92 −ζ9−4 −ζ9−1 −ζ9−2 −ζ94
162.14.3b6 C 3 −1 3ζ93 3ζ9−3 0 0 0 0 −ζ9−3 −ζ93 ζ9−ζ94 −2ζ9−4−ζ92 ζ9−4+2ζ92 −2ζ9−ζ94 ζ9+2ζ94 ζ9−4−ζ92 −ζ9−1 −ζ9−2 −ζ94 −ζ9 −ζ92 −ζ9−4
162.14.6a R 6 0 6 6 −3 0 0 0 0 0 0 0 0 0 0 0 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