Properties

Label 14T29
14T29 1 3 1->3 2 4 2->4 5 3->5 6 4->6 7 5->7 8 6->8 9 7->9 14 7->14 10 8->10 11 9->11 12 10->12 13 11->13 12->14 13->1 14->2
Degree $14$
Order $896$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2 \wr C_7$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(14, 29);
 
Copy content sage:G = TransitiveGroup(14, 29)
 
Copy content oscar:G = transitive_group(14, 29)
 
Copy content gap:G := TransitiveGroup(14, 29);
 

Group invariants

Abstract group:  $C_2 \wr C_7$
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:  $896=2^{7} \cdot 7$
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 $n$:  $14$
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 $t$:  $29$
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);
 
CHM label:   $[2^{7}]7=2wr7$
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);
 
$\card{\Aut(F/K)}$:  $2$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(14).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(14), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(14), G));
 
Generators:  $(7,14)$, $(1,3,5,7,9,11,13)(2,4,6,8,10,12,14)$
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)
$2$:  $C_2$
$7$:  $C_7$
$14$:  $C_{14}$
$56$:  $C_2^3:C_7$ x 2
$112$:  14T9 x 2
$448$:  14T21

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $C_7$

Low degree siblings

14T29 x 6, 28T104 x 7, 28T110 x 21, 28T111 x 42, 28T112 x 42, 28T113 x 21, 28T114 x 42, 28T115 x 42, 28T116 x 14, 28T117 x 42, 28T118 x 7

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{14}$ $1$ $1$ $0$ $()$
2A $2^{7}$ $1$ $2$ $7$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
2B $2^{6},1^{2}$ $7$ $2$ $6$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)$
2C $2^{2},1^{10}$ $7$ $2$ $2$ $( 2, 9)( 6,13)$
2D $2^{2},1^{10}$ $7$ $2$ $2$ $( 3,10)( 5,12)$
2E $2^{4},1^{6}$ $7$ $2$ $4$ $( 1, 8)( 3,10)( 6,13)( 7,14)$
2F $2^{2},1^{10}$ $7$ $2$ $2$ $( 4,11)( 5,12)$
2G $2^{4},1^{6}$ $7$ $2$ $4$ $( 1, 8)( 3,10)( 4,11)( 5,12)$
2H $2^{4},1^{6}$ $7$ $2$ $4$ $( 1, 8)( 2, 9)( 4,11)( 6,13)$
2I $2^{4},1^{6}$ $7$ $2$ $4$ $( 2, 9)( 3,10)( 5,12)( 6,13)$
2J $2^{4},1^{6}$ $7$ $2$ $4$ $( 2, 9)( 3,10)( 4,11)( 5,12)$
2K $2,1^{12}$ $7$ $2$ $1$ $( 7,14)$
2L $2^{5},1^{4}$ $7$ $2$ $5$ $( 1, 8)( 3,10)( 4,11)( 5,12)( 7,14)$
2M $2^{5},1^{4}$ $7$ $2$ $5$ $( 1, 8)( 2, 9)( 4,11)( 6,13)( 7,14)$
2N $2^{3},1^{8}$ $7$ $2$ $3$ $( 2, 9)( 4,11)( 5,12)$
2O $2^{5},1^{4}$ $7$ $2$ $5$ $( 1, 8)( 2, 9)( 3,10)( 6,13)( 7,14)$
2P $2^{3},1^{8}$ $7$ $2$ $3$ $( 2, 9)( 6,13)( 7,14)$
2Q $2^{3},1^{8}$ $7$ $2$ $3$ $( 3,10)( 5,12)( 7,14)$
2R $2^{3},1^{8}$ $7$ $2$ $3$ $( 1, 8)( 4,11)( 7,14)$
2S $2^{3},1^{8}$ $7$ $2$ $3$ $( 1, 8)( 6,13)( 7,14)$
7A1 $7^{2}$ $64$ $7$ $12$ $( 1,10,12, 7, 2, 4, 6)( 3, 5,14, 9,11,13, 8)$
7A-1 $7^{2}$ $64$ $7$ $12$ $( 1, 6, 4, 2, 7,12,10)( 3, 8,13,11, 9,14, 5)$
7A2 $7^{2}$ $64$ $7$ $12$ $( 1,12, 2, 6,10, 7, 4)( 3,14,11, 8, 5, 9,13)$
7A-2 $7^{2}$ $64$ $7$ $12$ $( 1, 4, 7,10, 6, 2,12)( 3,13, 9, 5, 8,11,14)$
7A3 $7^{2}$ $64$ $7$ $12$ $( 1, 7, 6,12, 4,10, 2)( 3, 9, 8,14,13, 5,11)$
7A-3 $7^{2}$ $64$ $7$ $12$ $( 1, 2,10, 4,12, 6, 7)( 3,11, 5,13,14, 8, 9)$
14A1 $14$ $64$ $14$ $13$ $( 1, 9,10,11,12,13, 7, 8, 2, 3, 4, 5, 6,14)$
14A-1 $14$ $64$ $14$ $13$ $( 1,14, 6, 5, 4, 3, 2, 8, 7,13,12,11,10, 9)$
14A3 $14$ $64$ $14$ $13$ $( 1,11, 7, 3, 6, 9,12, 8, 4,14,10,13, 2, 5)$
14A-3 $14$ $64$ $14$ $13$ $( 1, 5, 2,13,10,14, 4, 8,12, 9, 6, 3, 7,11)$
14A5 $14$ $64$ $14$ $13$ $( 1,13, 4, 9, 7, 5,10, 8, 6,11, 2,14,12, 3)$
14A-5 $14$ $64$ $14$ $13$ $( 1, 3,12,14, 2,11, 6, 8,10, 5, 7, 9, 4,13)$

Malle's constant $a(G)$:     $1$

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

32 x 32 character table

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