Properties

Label 27T415
27T415 1 15 1->15 18 1->18 2 13 2->13 17 2->17 3 3->15 16 3->16 4 14 4->14 4->14 5 10 5->10 5->18 6 11 6->11 12 6->12 7 7->10 7->16 8 8->12 8->17 9 9->11 9->13 19 10->19 22 10->22 26 11->26 27 11->27 12->19 20 12->20 24 13->24 13->24 14->22 23 14->23 21 15->21 25 15->25 16->20 16->23 17->25 17->27 18->21 18->26 19->6 19->6 20->1 20->2 21->8 21->9 22->1 22->4 23->3 23->8 24->7 24->9 25->2 25->5 26->3 26->7 27->4 27->5
Degree $27$
Order $1944$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_3^3:\PSU(3,2)$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(27, 415);
 
Copy content sage:G = TransitiveGroup(27, 415)
 
Copy content oscar:G = transitive_group(27, 415)
 
Copy content gap:G := TransitiveGroup(27, 415);
 

Group invariants

Abstract group:  $C_3^3:\PSU(3,2)$
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:  $1944=2^{3} \cdot 3^{5}$
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 $t$:  $415$
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:  $(1,18,26,3,15,21,9,13,24,7,16,20)(2,17,25)(4,14,23,8,12,19,6,11,27,5,10,22)$, $(1,15,25,5,18,21,8,17,27,4,14,22)(2,13,24,9,11,26,7,10,19,6,12,20)(3,16,23)$
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
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$8$:  $Q_8$
$12$:  $C_6\times C_2$
$24$:  24T4
$72$:  $C_3^2:Q_8$
$216$:  24T565

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 9: None

Low degree siblings

36T2813, 36T2814 x 2, 36T2994, 36T2995 x 2

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 $2^{12},1^{3}$ $81$ $2$ $12$ $( 1, 4)( 2, 3)( 5, 9)( 6, 8)(10,16)(11,15)(12,14)(17,18)(19,22)(20,21)(23,27)(24,26)$
3A $3^{9}$ $8$ $3$ $18$ $( 1, 3, 8)( 2, 4, 6)( 5, 7, 9)(10,17,12)(11,15,13)(14,18,16)(19,21,26)(20,22,24)(23,25,27)$
3B1 $3^{9}$ $9$ $3$ $18$ $( 1,10,19)( 2,18,20)( 3,17,21)( 4,16,22)( 5,15,23)( 6,14,24)( 7,13,25)( 8,12,26)( 9,11,27)$
3B-1 $3^{9}$ $9$ $3$ $18$ $( 1,19,10)( 2,20,18)( 3,21,17)( 4,22,16)( 5,23,15)( 6,24,14)( 7,25,13)( 8,26,12)( 9,27,11)$
3C $3^{6},1^{9}$ $24$ $3$ $12$ $( 1, 8, 3)( 2, 6, 4)( 5, 9, 7)(19,21,26)(20,22,24)(23,25,27)$
3D $3^{9}$ $24$ $3$ $18$ $( 1, 6, 5)( 2, 7, 3)( 4, 9, 8)(10,11,18)(12,13,14)(15,16,17)(19,21,26)(20,22,24)(23,25,27)$
3E $3^{9}$ $24$ $3$ $18$ $( 1, 5, 6)( 2, 3, 7)( 4, 8, 9)(10,16,13)(11,17,14)(12,18,15)(19,21,26)(20,22,24)(23,25,27)$
3F1 $3^{9}$ $72$ $3$ $18$ $( 1,17,24)( 2,16,25)( 3,12,20)( 4,14,27)( 5,13,19)( 6,18,23)( 7,11,21)( 8,10,22)( 9,15,26)$
3F-1 $3^{9}$ $72$ $3$ $18$ $( 1,24,17)( 2,25,16)( 3,20,12)( 4,27,14)( 5,19,13)( 6,23,18)( 7,21,11)( 8,22,10)( 9,26,15)$
4A $4^{6},1^{3}$ $162$ $4$ $18$ $( 1, 9, 4, 5)( 2, 8, 3, 6)(10,11,16,15)(12,17,14,18)(19,27,22,23)(20,26,21,24)$
4B $4^{6},1^{3}$ $162$ $4$ $18$ $( 1, 2, 6, 8)( 3, 9, 4, 7)(10,14,17,13)(11,18,16,15)(19,22,27,24)(21,23,25,26)$
4C $4^{6},1^{3}$ $162$ $4$ $18$ $( 1, 2, 5, 4)( 3, 7, 9, 8)(10,14,16,12)(11,17,15,18)(19,22,21,27)(20,25,23,24)$
6A1 $6^{4},3$ $81$ $6$ $22$ $( 1,22,10, 4,19,16)( 2,21,18, 3,20,17)( 5,27,15, 9,23,11)( 6,26,14, 8,24,12)( 7,25,13)$
6A-1 $6^{4},3$ $81$ $6$ $22$ $( 1,16,19, 4,10,22)( 2,17,20, 3,18,21)( 5,11,23, 9,15,27)( 6,12,24, 8,14,26)( 7,13,25)$
12A1 $12^{2},3$ $162$ $12$ $24$ $( 1,15,22, 9,10,23, 4,11,19, 5,16,27)( 2,14,21, 8,18,24, 3,12,20, 6,17,26)( 7,13,25)$
12A-1 $12^{2},3$ $162$ $12$ $24$ $( 1,27,16, 5,19,11, 4,23,10, 9,22,15)( 2,26,17, 6,20,12, 3,24,18, 8,21,14)( 7,25,13)$
12B1 $12^{2},3$ $162$ $12$ $24$ $( 1,23,11, 2,25,18, 6,26,16, 8,21,15)( 3,22,13, 9,27,10, 4,24,14, 7,19,17)( 5,20,12)$
12B-1 $12^{2},3$ $162$ $12$ $24$ $( 1,15,21, 8,16,26, 6,18,25, 2,11,23)( 3,17,19, 7,14,24, 4,10,27, 9,13,22)( 5,12,20)$
12C1 $12^{2},3$ $162$ $12$ $24$ $( 1,15,22, 2,18,21, 5,11,27, 4,17,19)( 3,14,20, 7,16,25, 9,12,23, 8,10,24)( 6,13,26)$
12C-1 $12^{2},3$ $162$ $12$ $24$ $( 1,19,17, 4,27,11, 5,21,18, 2,22,15)( 3,24,10, 8,23,12, 9,25,16, 7,20,14)( 6,26,13)$

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 3B1 3B-1 3C 3D 3E 3F1 3F-1 4A 4B 4C 6A1 6A-1 12A1 12A-1 12B1 12B-1 12C1 12C-1
Size 1 81 8 9 9 24 24 24 72 72 162 162 162 81 81 162 162 162 162 162 162
2 P 1A 1A 3A 3B-1 3B1 3C 3D 3E 3F-1 3F1 2A 2A 2A 3B1 3B-1 6A1 6A-1 6A-1 6A1 6A1 6A-1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 4A 4B 4C 2A 2A 4A 4A 4B 4B 4C 4C
Type
1944.2330.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1944.2330.1b R 1 1 1 1 1 1 1 1 1 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1
1944.2330.1c R 1 1 1 1 1 1 1 1 1 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1
1944.2330.1d R 1 1 1 1 1 1 1 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
1944.2330.1e1 C 1 1 1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
1944.2330.1e2 C 1 1 1 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
1944.2330.1f1 C 1 1 1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3
1944.2330.1f2 C 1 1 1 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1
1944.2330.1g1 C 1 1 1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 −1 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3
1944.2330.1g2 C 1 1 1 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 −1 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1
1944.2330.1h1 C 1 1 1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
1944.2330.1h2 C 1 1 1 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
1944.2330.2a S 2 −2 2 2 2 2 2 2 2 2 0 0 0 −2 −2 0 0 0 0 0 0
1944.2330.2b1 C 2 −2 2 2ζ3−1 2ζ3 2 2 2 2ζ3 2ζ3−1 0 0 0 −2ζ3 −2ζ3−1 0 0 0 0 0 0
1944.2330.2b2 C 2 −2 2 2ζ3 2ζ3−1 2 2 2 2ζ3−1 2ζ3 0 0 0 −2ζ3−1 −2ζ3 0 0 0 0 0 0
1944.2330.8a R 8 0 8 8 8 −1 −1 −1 −1 −1 0 0 0 0 0 0 0 0 0 0 0
1944.2330.8b1 C 8 0 8 8ζ3−1 8ζ3 −1 −1 −1 −ζ3 −ζ3−1 0 0 0 0 0 0 0 0 0 0 0
1944.2330.8b2 C 8 0 8 8ζ3 8ζ3−1 −1 −1 −1 −ζ3−1 −ζ3 0 0 0 0 0 0 0 0 0 0 0
1944.2330.24a R 24 0 −3 0 0 6 −3 −3 0 0 0 0 0 0 0 0 0 0 0 0 0
1944.2330.24b R 24 0 −3 0 0 −3 −3 6 0 0 0 0 0 0 0 0 0 0 0 0 0
1944.2330.24c R 24 0 −3 0 0 −3 6 −3 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