↑

Properties

Label 27T20
27T20 1 12 1->12 2 10 2->10 3 11 3->11 4 5 4->5 13 4->13 6 5->6 14 5->14 6->4 15 6->15 7 7->15 16 7->16 8 8->13 17 8->17 9 9->14 18 9->18 10->7 21 10->21 11->8 19 11->19 12->9 20 12->20 13->11 23 13->23 14->12 24 14->24 15->10 22 15->22 16->20 26 16->26 17->21 27 17->27 18->19 25 18->25 19->2 19->24 20->3 20->22 21->1 21->23 22->5 22->16 23->6 23->17 24->4 24->18 25->8 25->27 26->9 26->25 27->7 27->26
Degree $27$
Order $81$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_3.\He_3$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(27, 20);
 
Copy content sage:G = TransitiveGroup(27, 20)
 
Copy content oscar:G = transitive_group(27, 20)
 
Copy content gap:G := TransitiveGroup(27, 20);
 

Group invariants

Abstract group:  $C_3.\He_3$
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:  $81=3^{4}$
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:  $3$
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 $t$:  $20$
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);
 
$\card{\Aut(F/K)}$:  $3$
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:  $(4,5,6)(7,15,10)(8,13,11)(9,14,12)(16,20,22)(17,21,23)(18,19,24)(25,27,26)$, $(1,12,20,3,11,19,2,10,21)(4,13,23,6,15,22,5,14,24)(7,16,26,9,18,25,8,17,27)$
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)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$27$:  $C_3^2:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 9: $C_3^2:C_3$

Low degree siblings

27T26

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{27}$ $1$ $1$ $0$ $()$
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 $3^{9}$ $3$ $3$ $18$ $( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7,11,15)( 8,12,13)( 9,10,14)(16,19,22)(17,20,23)(18,21,24)$
3B-1 $3^{9}$ $3$ $3$ $18$ $( 1,25, 4)( 2,26, 5)( 3,27, 6)( 7,15,11)( 8,13,12)( 9,14,10)(16,22,19)(17,23,20)(18,24,21)$
3C1 $3^{8},1^{3}$ $9$ $3$ $16$ $( 1,26, 4)( 2,27, 5)( 3,25, 6)(10,12,11)(13,14,15)(16,20,23)(17,21,24)(18,19,22)$
3C-1 $3^{8},1^{3}$ $9$ $3$ $16$ $( 1, 4,26)( 2, 5,27)( 3, 6,25)(10,11,12)(13,15,14)(16,23,20)(17,24,21)(18,22,19)$
9A1 $9^{3}$ $3$ $9$ $24$ $( 1,12,20, 3,11,19, 2,10,21)( 4,13,23, 6,15,22, 5,14,24)( 7,16,26, 9,18,25, 8,17,27)$
9A-1 $9^{3}$ $3$ $9$ $24$ $( 1,23, 7, 2,24, 8, 3,22, 9)( 4,17,11, 5,18,12, 6,16,10)(13,27,19,14,25,20,15,26,21)$
9A2 $9^{3}$ $3$ $9$ $24$ $( 1,20,11, 2,21,12, 3,19,10)( 4,23,15, 5,24,13, 6,22,14)( 7,26,18, 8,27,16, 9,25,17)$
9A-2 $9^{3}$ $3$ $9$ $24$ $( 1,10,19, 3,12,21, 2,11,20)( 4,14,22, 6,13,24, 5,15,23)( 7,17,25, 9,16,27, 8,18,26)$
9A4 $9^{3}$ $3$ $9$ $24$ $( 1,11,21, 3,10,20, 2,12,19)( 4,15,24, 6,14,23, 5,13,22)( 7,18,27, 9,17,26, 8,16,25)$
9A-4 $9^{3}$ $3$ $9$ $24$ $( 1,19,12, 2,20,10, 3,21,11)( 4,22,13, 5,23,14, 6,24,15)( 7,25,16, 8,26,17, 9,27,18)$
9B1 $9^{3}$ $9$ $9$ $24$ $( 1, 9,18, 2, 7,16, 3, 8,17)( 4,12,19, 5,10,20, 6,11,21)(13,24,27,14,22,25,15,23,26)$
9B-1 $9^{3}$ $9$ $9$ $24$ $( 1,23,11, 3,22,10, 2,24,12)( 4,18,14, 6,17,13, 5,16,15)( 7,26,20, 9,25,19, 8,27,21)$
9C1 $9^{3}$ $9$ $9$ $24$ $( 1,18,10, 3,17,12, 2,16,11)( 4,20,15, 6,19,14, 5,21,13)( 7,26,23, 9,25,22, 8,27,24)$
9C-1 $9^{3}$ $9$ $9$ $24$ $( 1,13,22, 2,14,23, 3,15,24)( 4, 9,18, 5, 7,16, 6, 8,17)(10,19,25,11,20,26,12,21,27)$

Malle's constant $a(G)$:     $1/16$

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 3A1 3A-1 3B1 3B-1 3C1 3C-1 9A1 9A-1 9A2 9A-2 9A4 9A-4 9B1 9B-1 9C1 9C-1
Size 1 1 1 3 3 9 9 3 3 3 3 3 3 9 9 9 9
3 P 1A 3A-1 3A1 3B-1 3B1 3C-1 3C1 9A2 9A-2 9A4 9A-4 9A-1 9A1 9B-1 9B1 9C-1 9C1
Type
81.8.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
81.8.1b1 C 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1
81.8.1b2 C 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1
81.8.1c1 C 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3
81.8.1c2 C 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1
81.8.1d1 C 1 1 1 1 1 ζ3−1 ζ3 1 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1
81.8.1d2 C 1 1 1 1 1 ζ3 ζ3−1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3
81.8.1e1 C 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
81.8.1e2 C 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
81.8.3a1 C 3 3 3 3ζ3−1 3ζ3 0 0 0 0 0 0 0 0 0 0 0 0
81.8.3a2 C 3 3 3 3ζ3 3ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0
81.8.3b1 C 3 3ζ9−3 3ζ93 0 0 0 0 −ζ9+ζ94 −ζ9−4−2ζ92 2ζ9+ζ94 2ζ9−4+ζ92 −ζ9−4+ζ92 −ζ9−2ζ94 0 0 0 0
81.8.3b2 C 3 3ζ93 3ζ9−3 0 0 0 0 2ζ9−4+ζ92 2ζ9+ζ94 −ζ9−4−2ζ92 −ζ9+ζ94 −ζ9−2ζ94 −ζ9−4+ζ92 0 0 0 0
81.8.3b3 C 3 3ζ9−3 3ζ93 0 0 0 0 2ζ9+ζ94 −ζ9−4+ζ92 −ζ9−2ζ94 −ζ9−4−2ζ92 2ζ9−4+ζ92 −ζ9+ζ94 0 0 0 0
81.8.3b4 C 3 3ζ93 3ζ9−3 0 0 0 0 −ζ9−4−2ζ92 −ζ9−2ζ94 −ζ9−4+ζ92 2ζ9+ζ94 −ζ9+ζ94 2ζ9−4+ζ92 0 0 0 0
81.8.3b5 C 3 3ζ9−3 3ζ93 0 0 0 0 −ζ9−2ζ94 2ζ9−4+ζ92 −ζ9+ζ94 −ζ9−4+ζ92 −ζ9−4−2ζ92 2ζ9+ζ94 0 0 0 0
81.8.3b6 C 3 3ζ93 3ζ9−3 0 0 0 0 −ζ9−4+ζ92 −ζ9+ζ94 2ζ9−4+ζ92 −ζ9−2ζ94 2ζ9+ζ94 −ζ9−4−2ζ92 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