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Properties

Label 27T46
27T46 1 2 1->2 20 1->20 3 2->3 21 2->21 3->1 19 3->19 4 5 4->5 24 4->24 6 5->6 22 5->22 6->4 23 6->23 7 10 7->10 16 7->16 26 7->26 8 11 8->11 17 8->17 27 8->27 9 12 9->12 18 9->18 25 9->25 10->2 13 10->13 10->19 11->3 14 11->14 11->20 12->1 15 12->15 12->21 13->6 13->7 13->24 14->4 14->8 14->22 15->5 15->9 15->23 16->8 16->9 16->24 17->7 17->9 17->22 18->7 18->8 18->23 19->11 19->12 19->16 20->10 20->12 20->17 21->10 21->11 21->18 22->13 22->15 22->20 23->13 23->14 23->21 24->14 24->15 24->19 25->17 25->26 26->18 26->27 27->16 27->25
Degree $27$
Order $162$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_3^3:S_3$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(27, 46);
 
Copy content sage:G = TransitiveGroup(27, 46)
 
Copy content oscar:G = transitive_group(27, 46)
 
Copy content gap:G := TransitiveGroup(27, 46);
 

Group invariants

Abstract group:  $C_3^3:S_3$
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 $t$:  $46$
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:  $(1,20,12)(2,21,10)(3,19,11)(4,24,14)(5,22,15)(6,23,13)(7,26,18)(8,27,16)(9,25,17)$, $(1,2,3)(4,5,6)(7,16,9,18,8,17)(10,19,12,21,11,20)(13,24,15,23,14,22)(25,26,27)$, $(7,10,13)(8,11,14)(9,12,15)(16,24,19)(17,22,20)(18,23,21)$
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)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$ x 4, $C_6$
$18$:  $S_3\times C_3$ x 4, $C_3^2:C_2$
$54$:  $(C_3^2:C_3):C_2$ x 3, 18T23

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$, $S_3$

Degree 9: $S_3\times C_3$, $(C_3^2:C_3):C_2$ x 3

Low degree siblings

27T46 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 $1^{27}$ $1$ $1$ $0$ $()$
2A $2^{9},1^{9}$ $9$ $2$ $9$ $( 7,18)( 8,16)( 9,17)(10,21)(11,19)(12,20)(13,23)(14,24)(15,22)$
3A1 $3^{9}$ $1$ $3$ $18$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)$
3A-1 $3^{9}$ $1$ $3$ $18$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)$
3B1 $3^{9}$ $1$ $3$ $18$ $( 1, 4,26)( 2, 5,27)( 3, 6,25)( 7,12,14)( 8,10,15)( 9,11,13)(16,21,22)(17,19,23)(18,20,24)$
3B-1 $3^{9}$ $1$ $3$ $18$ $( 1,26, 4)( 2,27, 5)( 3,25, 6)( 7,14,12)( 8,15,10)( 9,13,11)(16,22,21)(17,23,19)(18,24,20)$
3C1 $3^{9}$ $1$ $3$ $18$ $( 1,27, 6)( 2,25, 4)( 3,26, 5)( 7,15,11)( 8,13,12)( 9,14,10)(16,23,20)(17,24,21)(18,22,19)$
3C-1 $3^{9}$ $1$ $3$ $18$ $( 1, 6,27)( 2, 4,25)( 3, 5,26)( 7,11,15)( 8,12,13)( 9,10,14)(16,20,23)(17,21,24)(18,19,22)$
3D1 $3^{9}$ $1$ $3$ $18$ $( 1, 5,25)( 2, 6,26)( 3, 4,27)( 7,10,13)( 8,11,14)( 9,12,15)(16,19,24)(17,20,22)(18,21,23)$
3D-1 $3^{9}$ $1$ $3$ $18$ $( 1,25, 5)( 2,26, 6)( 3,27, 4)( 7,13,10)( 8,14,11)( 9,15,12)(16,24,19)(17,22,20)(18,23,21)$
3E $3^{9}$ $6$ $3$ $18$ $( 1,22, 9)( 2,23, 7)( 3,24, 8)( 4,16,11)( 5,17,12)( 6,18,10)(13,26,21)(14,27,19)(15,25,20)$
3F $3^{6},1^{9}$ $6$ $3$ $12$ $( 7,13,10)( 8,14,11)( 9,15,12)(16,19,24)(17,20,22)(18,21,23)$
3G $3^{9}$ $6$ $3$ $18$ $( 1,22,12)( 2,23,10)( 3,24,11)( 4,16,14)( 5,17,15)( 6,18,13)( 7,26,21)( 8,27,19)( 9,25,20)$
3H $3^{9}$ $6$ $3$ $18$ $( 1, 9,20)( 2, 7,21)( 3, 8,19)( 4,11,24)( 5,12,22)( 6,10,23)(13,18,26)(14,16,27)(15,17,25)$
3I1 $3^{9}$ $6$ $3$ $18$ $( 1,24, 7)( 2,22, 8)( 3,23, 9)( 4,18,12)( 5,16,10)( 6,17,11)(13,25,19)(14,26,20)(15,27,21)$
3I-1 $3^{9}$ $6$ $3$ $18$ $( 1,23, 8)( 2,24, 9)( 3,22, 7)( 4,17,10)( 5,18,11)( 6,16,12)(13,27,20)(14,25,21)(15,26,19)$
3J1 $3^{9}$ $6$ $3$ $18$ $( 1, 3, 2)( 4, 6, 5)( 7,15,11)( 8,13,12)( 9,14,10)(16,21,22)(17,19,23)(18,20,24)(25,27,26)$
3J-1 $3^{9}$ $6$ $3$ $18$ $( 1, 2, 3)( 4, 5, 6)( 7,14,12)( 8,15,10)( 9,13,11)(16,20,23)(17,21,24)(18,19,22)(25,26,27)$
3K1 $3^{9}$ $6$ $3$ $18$ $( 1,24,10)( 2,22,11)( 3,23,12)( 4,18,15)( 5,16,13)( 6,17,14)( 7,25,19)( 8,26,20)( 9,27,21)$
3K-1 $3^{9}$ $6$ $3$ $18$ $( 1,23,11)( 2,24,12)( 3,22,10)( 4,17,13)( 5,18,14)( 6,16,15)( 7,27,20)( 8,25,21)( 9,26,19)$
3L1 $3^{9}$ $6$ $3$ $18$ $( 1, 8,21)( 2, 9,19)( 3, 7,20)( 4,10,22)( 5,11,23)( 6,12,24)(13,17,27)(14,18,25)(15,16,26)$
3L-1 $3^{9}$ $6$ $3$ $18$ $( 1, 7,19)( 2, 8,20)( 3, 9,21)( 4,12,23)( 5,10,24)( 6,11,22)(13,16,25)(14,17,26)(15,18,27)$
6A1 $6^{3},3^{3}$ $9$ $6$ $21$ $( 1, 2, 3)( 4, 5, 6)( 7,16, 9,18, 8,17)(10,19,12,21,11,20)(13,24,15,23,14,22)(25,26,27)$
6A-1 $6^{3},3^{3}$ $9$ $6$ $21$ $( 1, 3, 2)( 4, 6, 5)( 7,17, 8,18, 9,16)(10,20,11,21,12,19)(13,22,14,23,15,24)(25,27,26)$
6B1 $6^{3},3^{3}$ $9$ $6$ $21$ $( 1,26, 4)( 2,27, 5)( 3,25, 6)( 7,24,12,18,14,20)( 8,22,10,16,15,21)( 9,23,11,17,13,19)$
6B-1 $6^{3},3^{3}$ $9$ $6$ $21$ $( 1,24,26,20, 4,18)( 2,22,27,21, 5,16)( 3,23,25,19, 6,17)( 7,12,14)( 8,10,15)( 9,11,13)$
6C1 $6^{3},3^{3}$ $9$ $6$ $21$ $( 1,23,27,20, 6,16)( 2,24,25,21, 4,17)( 3,22,26,19, 5,18)( 7,11,15)( 8,12,13)( 9,10,14)$
6C-1 $6^{3},3^{3}$ $9$ $6$ $21$ $( 1,27, 6)( 2,25, 4)( 3,26, 5)( 7,22,11,18,15,19)( 8,23,12,16,13,20)( 9,24,10,17,14,21)$
6D1 $6^{3},3^{3}$ $9$ $6$ $21$ $( 1,25, 5)( 2,26, 6)( 3,27, 4)( 7,23,10,18,13,21)( 8,24,11,16,14,19)( 9,22,12,17,15,20)$
6D-1 $6^{3},3^{3}$ $9$ $6$ $21$ $( 1,22,25,20, 5,17)( 2,23,26,21, 6,18)( 3,24,27,19, 4,16)( 7,10,13)( 8,11,14)( 9,12,15)$

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 3B1 3B-1 3C1 3C-1 3D1 3D-1 3E 3F 3G 3H 3I1 3I-1 3J1 3J-1 3K1 3K-1 3L1 3L-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 6D1 6D-1
Size 1 9 1 1 1 1 1 1 1 1 6 6 6 6 6 6 6 6 6 6 6 6 9 9 9 9 9 9 9 9
2 P 1A 1A 3A-1 3A1 3B-1 3B1 3C-1 3C1 3D-1 3D1 3E 3F 3G 3H 3I-1 3I1 3J-1 3J1 3K-1 3K1 3L-1 3L1 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2A 2A
Type
162.41.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
162.41.1b R 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
162.41.1c1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 1 1 ζ3 1 1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3−1 1 ζ3 ζ3 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 1
162.41.1c2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 1 1 ζ3−1 1 1 ζ3−1 ζ3−1 ζ3 ζ3 ζ3 1 ζ3−1 ζ3−1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 1
162.41.1d1 C 1 −1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 1 1 ζ3 1 1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3−1 1 ζ3 ζ3 1 ζ3−1 −ζ3 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −1
162.41.1d2 C 1 −1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 1 1 ζ3−1 1 1 ζ3−1 ζ3−1 ζ3 ζ3 ζ3 1 ζ3−1 ζ3−1 1 ζ3 −ζ3−1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3 −1
162.41.2a R 2 0 2 2 2 2 2 2 2 2 −1 −1 −1 −1 −1 −1 −1 −1 −1 2 2 2 0 0 0 0 0 0 0 0
162.41.2b R 2 0 2 2 2 2 2 2 2 2 −1 −1 −1 −1 −1 2 −1 2 2 −1 −1 −1 0 0 0 0 0 0 0 0
162.41.2c R 2 0 2 2 2 2 2 2 2 2 −1 2 2 −1 −1 −1 2 −1 −1 −1 −1 −1 0 0 0 0 0 0 0 0
162.41.2d R 2 0 2 2 2 2 2 2 2 2 2 −1 −1 2 2 −1 −1 −1 −1 −1 −1 −1 0 0 0 0 0 0 0 0
162.41.2e1 C 2 0 2ζ3−1 2ζ3 2ζ3 2ζ3−1 2ζ3−1 2 2 2ζ3 −1 −1 −ζ3 −ζ3 −ζ3−1 2ζ3−1 −ζ3−1 2 2ζ3 −ζ3 −1 −ζ3−1 0 0 0 0 0 0 0 0
162.41.2e2 C 2 0 2ζ3 2ζ3−1 2ζ3−1 2ζ3 2ζ3 2 2 2ζ3−1 −1 −1 −ζ3−1 −ζ3−1 −ζ3 2ζ3 −ζ3 2 2ζ3−1 −ζ3−1 −1 −ζ3 0 0 0 0 0 0 0 0
162.41.2f1 C 2 0 2ζ3−1 2ζ3 2ζ3 2ζ3−1 2ζ3−1 2 2 2ζ3 −1 −1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3−1 −1 −ζ3 2ζ3 2 2ζ3−1 0 0 0 0 0 0 0 0
162.41.2f2 C 2 0 2ζ3 2ζ3−1 2ζ3−1 2ζ3 2ζ3 2 2 2ζ3−1 −1 −1 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3 −1 −ζ3−1 2ζ3−1 2 2ζ3 0 0 0 0 0 0 0 0
162.41.2g1 C 2 0 2ζ3−1 2ζ3 2ζ3 2ζ3−1 2ζ3−1 2 2 2ζ3 −1 2 2ζ3 −ζ3 −ζ3−1 −ζ3−1 2ζ3−1 −1 −ζ3 −ζ3 −1 −ζ3−1 0 0 0 0 0 0 0 0
162.41.2g2 C 2 0 2ζ3 2ζ3−1 2ζ3−1 2ζ3 2ζ3 2 2 2ζ3−1 −1 2 2ζ3−1 −ζ3−1 −ζ3 −ζ3 2ζ3 −1 −ζ3−1 −ζ3−1 −1 −ζ3 0 0 0 0 0 0 0 0
162.41.2h1 C 2 0 2ζ3−1 2ζ3 2ζ3 2ζ3−1 2ζ3−1 2 2 2ζ3 2 −1 −ζ3 2ζ3 2ζ3−1 −ζ3−1 −ζ3−1 −1 −ζ3 −ζ3 −1 −ζ3−1 0 0 0 0 0 0 0 0
162.41.2h2 C 2 0 2ζ3 2ζ3−1 2ζ3−1 2ζ3 2ζ3 2 2 2ζ3−1 2 −1 −ζ3−1 2ζ3−1 2ζ3 −ζ3 −ζ3 −1 −ζ3−1 −ζ3−1 −1 −ζ3 0 0 0 0 0 0 0 0
162.41.3a1 C 3 1 3ζ3−1 3ζ3 3ζ3−1 3 3ζ3 3ζ3 3ζ3−1 3 0 0 0 0 0 0 0 0 0 0 0 0 ζ3−1 ζ3 ζ3 1 ζ3−1 1 ζ3 ζ3−1
162.41.3a2 C 3 1 3ζ3 3ζ3−1 3ζ3 3 3ζ3−1 3ζ3−1 3ζ3 3 0 0 0 0 0 0 0 0 0 0 0 0 ζ3 ζ3−1 ζ3−1 1 ζ3 1 ζ3−1 ζ3
162.41.3b1 C 3 1 3ζ3−1 3ζ3 3 3ζ3 3 3ζ3−1 3ζ3 3ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 1 ζ3
162.41.3b2 C 3 1 3ζ3 3ζ3−1 3 3ζ3−1 3 3ζ3 3ζ3−1 3ζ3 0 0 0 0 0 0 0 0 0 0 0 0 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 1 ζ3−1
162.41.3c1 C 3 1 3 3 3ζ3−1 3ζ3−1 3ζ3 3ζ3−1 3ζ3 3ζ3 0 0 0 0 0 0 0 0 0 0 0 0 ζ3−1 ζ3−1 1 ζ3−1 1 ζ3 ζ3 ζ3
162.41.3c2 C 3 1 3 3 3ζ3 3ζ3 3ζ3−1 3ζ3 3ζ3−1 3ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 ζ3 ζ3 1 ζ3 1 ζ3−1 ζ3−1 ζ3−1
162.41.3d1 C 3 −1 3ζ3−1 3ζ3 3ζ3−1 3 3ζ3 3ζ3 3ζ3−1 3 0 0 0 0 0 0 0 0 0 0 0 0 −ζ3−1 −ζ3 −ζ3 −1 −ζ3−1 −1 −ζ3 −ζ3−1
162.41.3d2 C 3 −1 3ζ3 3ζ3−1 3ζ3 3 3ζ3−1 3ζ3−1 3ζ3 3 0 0 0 0 0 0 0 0 0 0 0 0 −ζ3 −ζ3−1 −ζ3−1 −1 −ζ3 −1 −ζ3−1 −ζ3
162.41.3e1 C 3 −1 3ζ3−1 3ζ3 3 3ζ3 3 3ζ3−1 3ζ3 3ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −1 −ζ3
162.41.3e2 C 3 −1 3ζ3 3ζ3−1 3 3ζ3−1 3 3ζ3 3ζ3−1 3ζ3 0 0 0 0 0 0 0 0 0 0 0 0 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −1 −ζ3−1
162.41.3f1 C 3 −1 3 3 3ζ3−1 3ζ3−1 3ζ3 3ζ3−1 3ζ3 3ζ3 0 0 0 0 0 0 0 0 0 0 0 0 −ζ3−1 −ζ3−1 −1 −ζ3−1 −1 −ζ3 −ζ3 −ζ3
162.41.3f2 C 3 −1 3 3 3ζ3 3ζ3 3ζ3−1 3ζ3 3ζ3−1 3ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 −ζ3 −ζ3 −1 −ζ3 −1 −ζ3−1 −ζ3−1 −ζ3−1

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