↑

Properties

Label 27T133
27T133 1 16 1->16 19 1->19 2 18 2->18 21 2->21 3 17 3->17 20 3->20 4 15 4->15 25 4->25 5 14 5->14 27 5->27 6 13 6->13 26 6->26 7 11 7->11 23 7->23 8 10 8->10 22 8->22 9 12 9->12 24 9->24 10->3 10->9 11->2 11->8 12->1 12->7 13->4 13->7 14->6 14->9 15->5 15->8 16->2 16->6 17->1 17->5 18->3 18->4 19->15 19->20 20->14 21->13 22->12 22->26 23->11 23->25 24->10 24->27 25->16 26->18 27->17
Degree $27$
Order $324$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_3^2.S_3^2$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(27, 133);
 
Copy content sage:G = TransitiveGroup(27, 133)
 
Copy content oscar:G = transitive_group(27, 133)
 
Copy content gap:G := TransitiveGroup(27, 133);
 

Group invariants

Abstract group:  $C_3^2.S_3^2$
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:  $324=2^{2} \cdot 3^{4}$
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 $t$:  $133$
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:  $(1,16,2,18,3,17)(4,15,5,14,6,13)(7,11,8,10,9,12)(19,20)(22,26)(23,25)(24,27)$, $(1,19,15,8,22,12)(2,21,13,7,23,11)(3,20,14,9,24,10)(4,25,16,6,26,18)(5,27,17)$
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)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$12$:  $D_{6}$ x 2
$36$:  $S_3^2$
$108$:  $C_3^2 : D_{6} $

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 2

Degree 9: $S_3^2$

Low degree siblings

27T123

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 $2^{9},1^{9}$ $9$ $2$ $9$ $( 1,23)( 2,24)( 3,22)( 4,27)( 5,25)( 6,26)( 7,19)( 8,20)( 9,21)$
2B $2^{13},1$ $27$ $2$ $13$ $( 1,13)( 2,15)( 3,14)( 4,10)( 5,12)( 6,11)( 7,18)( 8,17)( 9,16)(19,26)(20,25)(21,27)(23,24)$
2C $2^{12},1^{3}$ $27$ $2$ $12$ $( 1, 5)( 2, 4)( 3, 6)( 7, 8)(10,11)(13,17)(14,16)(15,18)(19,21)(22,27)(23,26)(24,25)$
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 $3^{6},1^{9}$ $6$ $3$ $12$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)$
3C $3^{9}$ $18$ $3$ $18$ $( 1,22,14)( 2,23,15)( 3,24,13)( 4,26,18)( 5,27,16)( 6,25,17)( 7,21,10)( 8,19,11)( 9,20,12)$
3D $3^{9}$ $36$ $3$ $18$ $( 1,19,16)( 2,20,17)( 3,21,18)( 4,22,10)( 5,23,11)( 6,24,12)( 7,27,14)( 8,25,15)( 9,26,13)$
6A $6^{3},3^{3}$ $18$ $6$ $21$ $( 1,22, 2,23, 3,24)( 4,26, 5,27, 6,25)( 7,21, 8,19, 9,20)(10,12,11)(13,15,14)(16,18,17)$
6B $6^{3},2^{4},1$ $54$ $6$ $19$ $( 1,14, 2,13, 3,15)( 4,11, 5,10, 6,12)( 7,16, 8,18, 9,17)(19,26)(20,25)(21,27)(23,24)$
6C $6^{4},3$ $54$ $6$ $22$ $( 1,16,22, 5,14,27)( 2,18,23, 4,15,26)( 3,17,24, 6,13,25)( 7,11,21, 8,10,19)( 9,12,20)$
9A1 $9^{3}$ $6$ $9$ $24$ $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,15,18,11,13,16,12,14,17)(19,22,26,20,23,27,21,24,25)$
9A2 $9^{3}$ $6$ $9$ $24$ $( 1, 9, 5, 3, 8, 4, 2, 7, 6)(10,18,13,12,17,15,11,16,14)(19,26,23,21,25,22,20,27,24)$
9A4 $9^{3}$ $6$ $9$ $24$ $( 1, 5, 8, 2, 6, 9, 3, 4, 7)(10,13,17,11,14,18,12,15,16)(19,23,25,20,24,26,21,22,27)$
18A1 $18,9$ $18$ $18$ $25$ $( 1,19, 4,22, 9,26, 2,20, 5,23, 7,27, 3,21, 6,24, 8,25)(10,16,15,12,18,14,11,17,13)$
18A5 $18,9$ $18$ $18$ $25$ $( 1,21, 5,22, 8,27, 2,19, 6,23, 9,25, 3,20, 4,24, 7,26)(10,18,13,12,17,15,11,16,14)$
18A7 $18,9$ $18$ $18$ $25$ $( 1,27, 9,24, 5,19, 3,26, 8,23, 4,21, 2,25, 7,22, 6,20)(10,15,18,11,13,16,12,14,17)$

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 2B 2C 3A 3B 3C 3D 6A 6B 6C 9A1 9A2 9A4 18A1 18A5 18A7
Size 1 9 27 27 2 6 18 36 18 54 54 6 6 6 18 18 18
2 P 1A 1A 1A 1A 3A 3B 3C 3D 3A 3B 3C 9A2 9A4 9A1 9A1 9A4 9A2
3 P 1A 2A 2B 2C 1A 1A 1A 1A 2A 2B 2C 3A 3A 3A 6A 6A 6A
Type
324.41.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
324.41.1b R 1 −1 −1 1 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1
324.41.1c R 1 −1 1 −1 1 1 1 1 −1 1 −1 1 1 1 −1 −1 −1
324.41.1d R 1 1 −1 −1 1 1 1 1 1 −1 −1 1 1 1 1 1 1
324.41.2a R 2 0 0 2 2 2 −1 −1 0 0 −1 2 2 2 0 0 0
324.41.2b R 2 2 0 0 2 2 2 −1 2 0 0 −1 −1 −1 −1 −1 −1
324.41.2c R 2 −2 0 0 2 2 2 −1 −2 0 0 −1 −1 −1 1 1 1
324.41.2d R 2 0 0 −2 2 2 −1 −1 0 0 1 2 2 2 0 0 0
324.41.4a R 4 0 0 0 4 4 −2 1 0 0 0 −2 −2 −2 0 0 0
324.41.6a R 6 0 2 0 6 −3 0 0 0 −1 0 0 0 0 0 0 0
324.41.6b R 6 0 −2 0 6 −3 0 0 0 1 0 0 0 0 0 0 0
324.41.6c1 R 6 2 0 0 −3 0 0 0 −1 0 0 ζ9−4+ζ9−ζ92+2ζ94 −2ζ9−4+ζ9−ζ92−ζ94 ζ9−4−2ζ9+2ζ92−ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9
324.41.6c2 R 6 2 0 0 −3 0 0 0 −1 0 0 ζ9−4−2ζ9+2ζ92−ζ94 ζ9−4+ζ9−ζ92+2ζ94 −2ζ9−4+ζ9−ζ92−ζ94 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94
324.41.6c3 R 6 2 0 0 −3 0 0 0 −1 0 0 −2ζ9−4+ζ9−ζ92−ζ94 ζ9−4−2ζ9+2ζ92−ζ94 ζ9−4+ζ9−ζ92+2ζ94 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92
324.41.6d1 R 6 −2 0 0 −3 0 0 0 1 0 0 ζ9−4+ζ9−ζ92+2ζ94 −2ζ9−4+ζ9−ζ92−ζ94 ζ9−4−2ζ9+2ζ92−ζ94 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9
324.41.6d2 R 6 −2 0 0 −3 0 0 0 1 0 0 ζ9−4−2ζ9+2ζ92−ζ94 ζ9−4+ζ9−ζ92+2ζ94 −2ζ9−4+ζ9−ζ92−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94
324.41.6d3 R 6 −2 0 0 −3 0 0 0 1 0 0 −2ζ9−4+ζ9−ζ92−ζ94 ζ9−4−2ζ9+2ζ92−ζ94 ζ9−4+ζ9−ζ92+2ζ94 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92

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